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## Jacques Duparc

### Coordonnées

 Professeur ordinaire ContactJacques.Duparc@unil.chInternef, bureau 134Tél 021.692.35.88Adresse postaleUniversité de LausanneQuartier UNIL-DorignyBâtiment Internef1015 Lausanne

### Enseignements

 bachelor Introduction à la logique Formation concernée Baccalauréat universitaire ès Sciences en management

### Axes de recherche

AutoMathA
De la théorie des automates des mathématiques aux applications
Pour plus d'information : www.esf.org/automatha

Complexité Topologique, Jeux, Logique et Automates
L'un des projets consiste à déméler la fine structure topologique des omega-langages d'arbres réguliers. En d'autres termes, il s'agit de découvrir la hierarchie de Wadge des automates d'arbres infinis non déterministes
L'autre cherche à caractériser les fonctions boréliennes comme des stratégies dans des jeux correctement définis. Le cas des fonctions de classe de Baire un entier fini fournissant une première étape déjà difficile

GAMES
Mise en oeuvre des méthodes formelles qui garantissent la fiabilité, la rectitude et l'efficacité des systèmes informatiques, en dévellopant des méthodes de spécification et de validation qui sont basées sur les jeux et les automates

Topologies profinies en théorie des automates
Etude des relations de réduction entre langages réguliers, basées sur fonctions uniformément continues - pour diverses topologies profinies.

### Compétences

Logique mathématique
Fondement des mathématiques (théorie des ensembles) et fondement de l'informatique (informatique théorique).

### Evénements

#### Conferences

CSL'07
Computer Science Logic (CSL) is the annual conference of the European Association for Computer Science Logic (EACSL)<br />
The conference is intended for computer scientists whose research activities involve logic, as well as for logicians working on issues significant for computer science <br />
CSL'07, the 16th annual EACSL conference will be organized in Lausanne by the Western Swiss Center for Logic, History and Philosophy of Sciences, and the University of Lausanne<br />
The Ackermann Award for 2007 is sponsored by Logitech and will be presented to the recipients at CSL'07<br />
<br />
A joint session with GAMES 07, the annual meeting of the European Network will take place on 11 September, 2007
Suisse

GAMES'07
The Annual GAMES-Meeting 2007 will be held in Lausanne, Switzerland, on September 10-13, 2007. It will be co-located with CSL 2007, with a joint programme on September 11. As in previous years, GAMES 2007 will be an informal workshop, without proceedings, with a programme consisting of five invited tutorials (90 min), contributed talks (30 min) and short presentations (15 min). Contributed talks and short presentations will be selected by the programme committee on the basis of submitted abstracts<br />
<br />
<br />

Suisse

### Publications

37 publications classées par: type de publication  -  année

: Revue avec comité de lecture

### Articles

 Jacques Duparc (2015). Easy Proofs of Löwenheim-Skolem Theorems by Means of Evaluation Games. CoRR, abs/1507.03665. [url] [abstract]AbstractWe propose a proof of the downward L\"owenheim-Skolem that relies on strategies deriving from evaluation games instead of the Skolem normal forms. This proof is simpler, and easily understood by the students, although it requires, when defining the semantics of first-order logic to introduce first a few notions inherited from game theory such as the one of an evaluation game.

 Duparc J. , Finkel O. (2007). Wadge games and sets of reals recognized by simple machines: An omega power of a finite context-free language which is Borel above Delta^ø_omega. Studies in Logic, 11, 109-122.

 Duparc J. , Riss M. (2006). The Missing Link for omega-Rational Sets, Automata, and Semigroups. International Journal of Algebra and Computation, 16, 161-186. [abstract]AbstractIn 1997, following the works of Klaus W. Wagner on omega-regular sets, Olivier Carton and Dominique Perrin introduced the notions of Chains and Superchains for omega-semigroups. There is a clear correspondance between the algebraic representation of each of these operations and the automata-theoretical one. Unfortunately, chains and superchains do not suffice to describe the whole Wagner hierarchy. We introduce a third notion which completes the task undertaken in Carton O., Perrin D.: "Chains and superchains for omega-rational sets, automata and semigroups" (Int. J. Alg. Comput., vol. 7, no. 7, pp. 673-695, 1997).

 Duparc J. (2003). A Hierarchy of Deterministic Context-Free omega-languages. Theoretical Computer Science, 290, 1253-1300. [abstract]AbstractTwenty years ago, Klaus. W. Wagner came up with a hierarchy of omega-regular sets that actually bears his name. It turned out to be exactly the Wadge hierarchy of the sets of omega-words recognized by deterministic finite automata. We describe the Wadge hierarchy of context-free omega-languages, which stands as an extension of Wagner's work from automata to pushdown automata.

 Duparc J. (2003). The Steel Hierarchy of Ordinal Valued Borel Mappings. Journal of Symbolic Logic, 68, 187-234. [url] [abstract]AbstractGiven well ordered countable sets of the form $\lamphi$, we consider Borel mappings from $\lamphiom$ with countable image inside the ordinals. The ordinals and $\lamphiom$ are respectively equipped with the discrete topology and the product of the discrete topology on $\lamphi$. The Steel well-ordering on such mappings is defined by $\phi\minf\psi$ iff there exists a continuous function $f$ such that $\phi(x)\leq\psi\circ f(x)$ holds for any $x\in\lamphiom$. It induces a hierarchy of mappings which we give a complete description of. We provide, for each ordinal $\alpha$, a mapping $\T\alpha$ whose rank is precisely $\alpha$ in this hierarchy and we also compute the height of the hierarchy restricted to mappings with image bounded by $\alpha$. These mappings being viewed as partitions of the reals, there is, in most cases, a unique distinguished element of the partition. We analyze the relation between its topological complexity and the rank of the mapping in this hierarchy.

 Duparc J. (2001). Wadge Hierarchy and Veblen Hierarchy. Part I: Borel Sets of Finite Rank. Journal of Symbolic Logic, 66, 56-86.

 Duparc J., Finkel O. ; Ressayre J-P. (2001). Computer Science and the Fine Structure of Borel Sets. Theoretical Computer Science, 257, 85-105. [abstract]AbstractI) Wadge defined a natural refinement of the Borel hierarchy, now called the Wadge hierarchy WH. The fundamental properties of WH follow from results of Kuratowski, Martin, Wadge and Louveau. We give a transparent restatement and proof of Wadge's main theorem. Our method is new for it yields a wide and unexpected extension: from Borel sets of reals to a class of natural but non Borel sets of infinite sequences. Wadge's theorem is quite uneffective and our generalization clearly worse in this respect. Yet paradoxically our method is appropriate to effectivize this whole theory in the context discussed below. II) Wagner defined on Büchi automata (accepting words of length ω) a hierarchy and proved for it an effective analog of Wadge's results. We extend Wagner's results to more general kinds of Automata: Counters, Push Down Automata and Büchi Automata reading transfinite words. The notions and methods developed in (I) are quite useful for this extension. And we start to use them in order to look for extensions of the fundamental effective determinacy results of Büchi-Landweber, Rabin; and of Courcelle-Walukiewicz.

 Duparc J. (1999). The Normal Form of Borel Sets. Part II: Borel Sets of Infinite Rank. C. R. Acad. Sci. Paris, t. 328, 735-740. [abstract]AbstractFor each Borel set of reals of infinite rank A we obtain a normal form'' of A by finding a Borel set Ω such that A and Ω continuously reduce to each other. We do so by defining simple Borel operations which are homomorphic to the ω1 first Veblen ordinal operations of base ω1 required to compute the Wadge degree of a Borel set.

 Duparc J. (1995). The Normal Form of Borel Sets. Part I: Borel Sets of Finite Rank. C. R. Acad. Sci. Paris, t. 320, p. 651-656. [abstract]AbstractFor each Borel set of reals A, of finite rank, we obtain a "normal form" of A, by finding a Borel set Ω of maximum simplicity, such that A and Ω continuously reduce to each other. In more technical terms : we define simple Borel operations which are homomorphic to ordinal sum, to multiplication by a countable ordinal, and to ordinal exponentiation of base ω1, under the map which sends every Borel set A of finite rank to its Wadge degree.

### Parties de livre

 Duparc J., Finkel O. ; Ressayre J-P. (2014). The Wadge Hierarchy of Petri Nets omega-Languages. Logic, Computation, Hierarchies (Vol. 4, pp. 109-138). De Gruyter. [doi] [abstract]AbstractWe describe the Wadge hierarchy of the omega-languages recognized by deterministic Petri nets. This is an extension of the celebrated Wagner hierarchy which turned out to be the Wadge hierarchy of the omega-regular languages. Petri nets are more powerful devices than finite automata. They may be defined as partially blind multi-counter automata. We show that the whole hierarchy has height omega(omega 2), and give a description of the restrictions of this hierarchy to partially blind multi-counter automata of some fixed positive number of counters.

 Arnold A., Duparc J., Murlak F. ; Niwiński D. (2007). On the topological complexity of tree languages. Logic and Automata: History and Perspectives (Vol. 2, pp. 9-28). Amsterdam University Press. [url] [abstract]AbstractThe article surveys recent results in the study of topological complexity of recognizable tree languages. Emphasis is put on the relation between topological hierarchies, like the Borel hierarchy or the Wadge hierarchy, and the hierarchies resulting from the structure of automata, as the Rabin-Mostowski index hierarchy. The topological complexity of recognizable tree languages is seen as an evidenceof their structural complexity, which also induces the computational complexity of the verification problems related to automata, as the non-emptiness problem. Indeed, the topological aspect can be seen as a rudiment of the infinite computation complexity theory.

### Actes de conférence (partie)

 Duparc J., Finkel O. ; Ressayre J.-P. (2013, Jan). The Wadge Hierarchy of Petri Nets \emph$$\omega$$-Languages. Logical Foundations of Computer Science, International Symposium, LFCS 2013, San Diego, CA, USA, January 6-8, 2013. Proceedings, 7734 (pp. 179-193). [doi] [url] [abstract]AbstractWe describe the Wadge hierarchy of the ω-languages recognized by deterministic Petri nets. This is an extension of the celebrated Wagner hierarchy which turned out to be the Wadge hierarchy of the ω-regular languages. Petri nets are an improvement of automata. They may be defined as partially blind multi-counter automata. We show that the whole hierarchy has height ωω2, and give a description of the restrictions of this hierarchy to every fixed number of partially blind counters.

 Duparc J., Facchini A. ; Murlak F. (2011, Jan). Definable Operations On Weakly Recognizable Sets of Trees. IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2011, December 12-14, 2011, Mumbai, India (pp. 363-374). [doi] [url] [abstract]AbstractAlternating automata on infinite trees induce operations on languages which do not preserve natural equivalence relations, like having the same Mostowski--Rabin index, the same Borel rank, or being continuously reducible to each other (Wadge equivalence). In order to prevent this, alternation needs to be restricted to the choice of direction in the tree. For weak alternating automata with restricted alternation a small set of computable operations generates all definable operations, which implies that the Wadge degree of a given automaton is computable. The weak index and the Borel rank coincide, and are computable. An equivalent automaton of minimal index can be computed in polynomial time (if the productive states of the automaton are given).

 Cabessa J., Duparc J., Facchini A. ; Murlak F. (2009, Déc). The Wadge Hierarchy of Max-Regular Languages. IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2009), 4 (pp. 121-132). Schloss Dagstuhl-Leibniz-Zentrum für Informatik. [doi] [url]

 Duparc J., Facchini A. ; Murlak F. (2009, Jan). Linear Game Automata: Decidable Hierarchy Problems for Stripped-Down Alternating Tree Automata. Computer Science Logic: 23rd International Workshop, CSL 2009, 18th Annual Conference of the EACSL, Coimbra, Portugal, September 7-11, 2009, Proceedings, 5771 (pp. 225-239). Springer. [url]

 Facchini A. , Duparc J. (2009, Jan). A Playful Glance at Hierarchical Questions for Two-Way Alternating Automata. Selected Papers of the International Conference Infinity in Logic and Computation, Cape Town, South Africa, November 2007. Springer.

 Duparc J. , Facchini A. (2008, Jan). Describing the Wadge Hierarchy for the Alternation Free Fragment of μ -Calculus (I) The Levels Below ω 1. Logic and Theory of Algorithms, Fourth Conference on Computability in Europe, CiE 2008, Athens, Greece, June 2008, Proceedings, 5028. Springer.

 Cabessa J. , Duparc J. (2007, Jan). An inﬁnite game on omega-semigroups. Inﬁnite Games, Papers of the conference Foundations of the Formal Sciences V, held in Bonn, November 26-29, 2004, 11 (pp. 63-78). Bold S. Löwe B. Räsch T. van Benthem J.

 Duparc J. , Finkel O. (2007, Jan). An ω-power of a ﬁnite context-free language which is Borel above ∆0ω. Inﬁnite Games, Papers of the conference Foundations of the Formal Sciences V, held in Bonn, November 26-29, 2004, 11 (pp. 109-122). Bold S. Löwe B. Räsch T. van Benthem J.

 Duparc J. , Henzinger T.A. (2007, Sep). Computer Science Logic. 21st International Workshop, CSL 2007 16th Annual Conference of the EACSL Lausanne, Switzerland, 4646. Springer.

 Duparc J. , Murlak F. (2007, Jan). On the Topological Complexity of Weakly Recognizable Tree Languages. Proceedings of the 16th International Symposium, Funda,mentals of Computation Theory, 4639 (pp. 261-273).

 Duparc J., Bradfield J. ; Quickert S. (2005, Jan). Transfinite extension ot the mu-calculus. Proceedings 14th annual European conference Computer Science Logic, 3634. Springer Berlin / Heidelberg.

 Duparc J. (2003, Jan). Posivitive Games and persistent Strategies. 12th Annual Conference of the European Association for Computer Science Logic, 2803 (pp. 182-196). Springer.

 Cachat T., Duparc J. ; Thomas W. (2002, Jan). Solving Pushdown Games with a Sigma_3 Winning Condition. Proceedings of the 11th Annual Conference of the European Association for Computer Science Logic, CSL 2002, 2471 (pp. 322-336).

 Duparc J., Finkel O. ; Ressayre J-P. (2000, Jan). Infinite Games, Finite Machines. Proceedings of the Joint Conference of the 5th Barcelona Logic Meeting and the 6th Kurt Gödel colloquium, Collegium Logicum, Annals of the Kurt-Gödel-Society Vol 4, 2000.

 Duparc J. (1994, Jan). The Normal Form of Borel Sets of Finite Rank. Contibuted Papers of the Logic Colloquium'94.

### Rapports

 Duparc J. , Facchini A. (2009). The Topological Complexity of Models of the Modal μ-Calculus: On The Alternation Free Fragment and Beyond. Laboratoire Bordelais de Recherche en Informatique et Université de Lausanne. [url]

### Thèses

 Fournier K., Duparc J. (Dir.) (2016). The Wadge Hierarchy: Beyond Borel Sets. Université de Lausanne, Faculté des hautes études commerciales.

 Carroy R., Duparc J. , Finkel O. (Dir.) (2013). Fonctions de première classe de Baire. Université de Lausanne, Faculté des hautes études commerciales.

 Bach C. W., Jacques D. , Hans P. (Dir.) (2010). Interactive Epistemology and Reasoning: On the Foundations of Game Theory. Université de Lausanne, Faculté des hautes études commerciales. [abstract]AbstractGame theory describes and analyzes strategic interaction. It is usually distinguished between static games, which are strategic situations in which the players choose only once as well as simultaneously, and dynamic games, which are strategic situations involving sequential choices. In addition, dynamic games can be further classified according to perfect and imperfect information. Indeed, a dynamic game is said to exhibit perfect information, whenever at any point of the game every player has full informational access to all choices that have been conducted so far. However, in the case of imperfect information some players are not fully informed about some choices. Game-theoretic analysis proceeds in two steps. Firstly, games are modelled by so-called form structures which extract and formalize the significant parts of the underlying strategic interaction. The basic and most commonly used models of games are the normal form, which rather sparsely describes a game merely in terms of the players' strategy sets and utilities, and the extensive form, which models a game in a more detailed way as a tree. In fact, it is standard to formalize static games with the normal form and dynamic games with the extensive form. Secondly, solution concepts are developed to solve models of games in the sense of identifying the choices that should be taken by rational players. Indeed, the ultimate objective of the classical approach to game theory, which is of normative character, is the development of a solution concept that is capable of identifying a unique choice for every player in an arbitrary game. However, given the large variety of games, it is not at all certain whether it is possible to device a solution concept with such universal capability. Alternatively, interactive epistemology provides an epistemic approach to game theory of descriptive character. This rather recent discipline analyzes the relation between knowledge, belief and choice of game-playing agents in an epistemic framework. The description of the players' choices in a given game relative to various epistemic assumptions constitutes the fundamental problem addressed by an epistemic approach to game theory. In a general sense, the objective of interactive epistemology consists in characterizing existing game-theoretic solution concepts in terms of epistemic assumptions as well as in proposing novel solution concepts by studying the game-theoretic implications of refined or new epistemic hypotheses. Intuitively, an epistemic model of a game can be interpreted as representing the reasoning of the players. Indeed, before making a decision in a game, the players reason about the game and their respective opponents, given their knowledge and beliefs. Precisely these epistemic mental states on which players base their decisions are explicitly expressible in an epistemic framework. In this PhD thesis, we consider an epistemic approach to game theory from a foundational point of view. In Chapter 1, basic game-theoretic notions as well as Aumann's epistemic framework for games are expounded and illustrated. Also, Aumann's sufficient conditions for backward induction are presented and his conceptual views discussed. In Chapter 2, Aumann's interactive epistemology is conceptually analyzed. In Chapter 3, which is based on joint work with Conrad Heilmann, a three-stage account for dynamic games is introduced and a type-based epistemic model is extended with a notion of agent connectedness. Then, sufficient conditions for backward induction are derived. In Chapter 4, which is based on joint work with Jérémie Cabessa, a topological approach to interactive epistemology is initiated. In particular, the epistemic-topological operator limit knowledge is defined and some implications for games considered. In Chapter 5, which is based on joint work with Jérémie Cabessa and Andrés Perea, Aumann's impossibility theorem on agreeing to disagree is revisited and weakened in the sense that possible contexts are provided in which agents can indeed agree to disagree.

 Facchini A., Duparc J. (Dir.) (2010). A study on the expressive power of some fragments of the modal µ-Calculus. Université de Lausanne, Faculté des hautes études commerciales. [abstract]AbstractRésumé Le μ-calcul est une extension de la logique modale par des opérateurs de point fixe. Dans ce travail nous étudions la complexité de certains fragments de cette logique selon deux points de vue, différents mais étroitement liés: l'un syntaxique (ou combinatoire) et l'autre topologique. Du point de vue syn¬taxique, les propriétés définissables dans ce formalisme sont classifiées selon la complexité combinatoire des formules de cette logique, c'est-à-dire selon le nombre d'alternances des opérateurs de point fixe. Comparer deux ensembles de modèles revient ainsi à comparer la complexité syntaxique des formules as¬sociées. Du point de vue topologique, les propriétés définissables dans cette logique sont comparées à l'aide de réductions continues ou selon leurs positions dans la hiérarchie de Borel ou dans celle projective. Dans la première partie de ce travail nous adoptons le point de vue syntax¬ique afin d'étudier le comportement du μ-calcul sur des classes restreintes de modèles. En particulier nous montrons que: (1) sur la classe des modèles symétriques et transitifs le μ-calcul est aussi expressif que la logique modale; (2) sur la classe des modèles transitifs, toute propriété définissable par une formule du μ-calcul est définissable par une formule sans alternance de points fixes, (3) sur la classe des modèles réflexifs, il y a pour tout η une propriété qui ne peut être définie que par une formule du μ-calcul ayant au moins η alternances de points fixes, (4) sur la classe des modèles bien fondés et transitifs le μ-calcul est aussi expressif que la logique modale. Le fait que le μ-calcul soit aussi expressif que la logique modale sur la classe des modèles bien fondés et transitifs est bien connu. Ce résultat est en ef¬fet la conséquence d'un théorème de point fixe prouvé indépendamment par De Jongh et Sambin au milieu des années 70. La preuve que nous donnons de l'effondrement de l'expressivité du μ-calcul sur cette classe de modèles est néanmoins indépendante de ce résultat. Par la suite, nous étendons le langage du μ-calcul en permettant aux opérateurs de point fixe de lier des occurrences négatives de variables libres. En montrant alors que ce formalisme est aussi ex¬pressif que le fragment modal, nous sommes en mesure de fournir une nouvelle preuve du théorème d'unicité des point fixes de Bernardi, De Jongh et Sambin et une preuve constructive du théorème d'existence de De Jongh et Sambin. RÉSUMÉ Pour ce qui concerne les modèles transitifs, du point de vue topologique cette fois, nous prouvons que la logique modale correspond au fragment borélien du μ-calcul sur cette classe des systèmes de transition. Autrement dit, nous vérifions que toute propriété définissable des modèles transitifs qui, du point de vue topologique, est une propriété borélienne, est nécessairement une propriété modale, et inversement. Cette caractérisation du fragment modal découle du fait que nous sommes en mesure de montrer que, modulo EF-bisimulation, un ensemble d'arbres est définissable dans la logique temporelle Ε F si et seulement il est borélien. Puisqu'il est possible de montrer que ces deux propriétés coïncident avec une caractérisation effective de la définissabilité dans la logique Ε F dans le cas des arbres à branchement fini donnée par Bojanczyk et Idziaszek [24], nous obtenons comme corollaire leur décidabilité. Dans une deuxième partie, nous étudions la complexité topologique d'un sous-fragment du fragment sans alternance de points fixes du μ-calcul. Nous montrons qu'un ensemble d'arbres est définissable par une formule de ce frag¬ment ayant au moins η alternances si et seulement si cette propriété se trouve au moins au n-ième niveau de la hiérarchie de Borel. Autrement dit, nous vérifions que pour ce fragment du μ-calcul, les points de vue topologique et combina- toire coïncident. De plus, nous décrivons une procédure effective capable de calculer pour toute propriété définissable dans ce langage sa position dans la hiérarchie de Borel, et donc le nombre d'alternances de points fixes nécessaires à la définir. Nous nous intéressons ensuite à la classification des ensembles d'arbres par réduction continue, et donnons une description effective de l'ordre de Wadge de la classe des ensembles d'arbres définissables dans le formalisme considéré. En particulier, la hiérarchie que nous obtenons a une hauteur (ωω)ω. Nous complétons ces résultats en décrivant un algorithme permettant de calculer la position dans cette hiérarchie de toute propriété définissable.

 Duparc J. (1995). La Forme Normale des Boréliens de Rang fini. University Paris 7.

### Non publié

 Duparc J. (In Press). A Normal Form for Borel Sets. [abstract]AbstractFor each Borel set A of reals, or more generally of infinite strings from a countable alphabet, we obtain a normal form'' ofA, by finding a Borel setΩ of maximum simplicity, such that A and Ω continuously reduce to each other.Ω only depend on the equivalence class of A modulo inter-reducibility, moreover, the map: A→Ω is defined in a simple way (in ZF+DC). In case of Borel sets of finite rank, we prove the above result essentially by defining simple Borel operations which are homomorphic to ordinal sum, to multiplication by a countable ordinal, and to ordinal exponentiation of base ω1 (under the map which sends every Borel setA of finite rank to its Wadge degree). Extension to transfinite ranks is provided by iterating both ordinal exponentiation and its Borel counterpart. And finally, we show that all these results can be extended to all Borel subsets of Xω when the alphabet X is of any cardinal.

 Duparc J. (In Press). Wadge Degrees of Louveau's (Wadge) Classes. [abstract]AbstractWe give for, each Louveau's class Γu, a Γu-complete set Au, that is [Aub] =Γu , in our terminology. This allows to compute the level of this Wadge class in the Wadge hierarchy.

 Duparc J. (In Press). Wadge Hierarchy and Veblen Hierarchy. Part II: Borel Sets of Infinite Rank. [abstract]AbstractWe consider Borel sets of the form A ⊆ Λω (with usual topology) where cardinality of Λ is less than some uncountable regular cardinal Κ. We obtain a normal form'' of A, by finding a Borel set Ω(α) such that A and Ω(α) continuously reduce to each other. We do so by defining Borel operations which are homomorphic to the Κ first Veblen ordinal functions of base Κ required to compute the Wadge degree of the set A: the ordinal α.

 Duparc J. (2008). A Normal Form of Borel Sets of Finite Rank. [abstract]AbstractFor each Borel set of reals A, of finite rank, we obtain a normal form'' of A, by finding a canonical Borel set Ω, such that A and Ω continuously reduce to each other. In more technical terms: we define simple Borel operations which are homomorphic to ordinal sum, to multiplication by a countable ordinal, and to ordinal exponentiation of base ω1 , under the map which sends every Borel set A of finite rank to its Wadge degree.

### Autres

 Duparc J. , Cabessa J. (2004). Games on Semigroups.

### Formations

Doctorat de mathématiques
Université Paris VII-Denis Diderot, (1995)

### Mots-clés

• automatha
• informatique théorique
• logique mathématique
• systèmes d'information (8)
• théorie descriptive des ensembles

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