From Normal Functors to Logarithmic Space Queries

We introduce a new approach to implicit complexity in linear logic, inspired by functional database query languages and using recent developments in effective denotational semantics of polymorphism. We give the first sub-polynomial upper bound in a type system with impredicative polymorphism; adding restrictions on quantifiers yields a characterization of logarithmic space, for which extensional completeness is established via descriptive complexity

On dialogue games and coherent strategies

We explain how to see the set of positions of a dialogue game as a coherence space in the sense of Girard or as a bistructure in the sense of Curien, Plotkin and Winskel. The coherence structure on the set of positions results from a Kripke translation of tensorial logic into linear logic extended with a necessity modality. The translation is done in such a way that every innocent strategy defines a clique or a configuration in the resulting space of positions. This leads us to study the notion of configuration designed by Curien, Plotkin and Winskel for general bistructures in the particular case of a bistructure associated to a dialogue game. We show that every such configuration may be seen as an interactive strategy equipped with a backward as well as a forward dynamics based on the interplay between the stable order and the extensional order. In that way, the category of bistructures is shown to include a full subcategory of games and coherent strategies of an interesting nature

Non uniform (hyper/multi)coherence spaces

In (hyper)coherence semantics, proofs/terms are cliques in (hyper)graphs. Intuitively, vertices represent results of computations and the edge relation witnesses the ability of being assembled into a same piece of data or a same (strongly) stable function, at arrow types. In (hyper)coherence semantics, the argument of a (strongly) stable functional is always a (strongly) stable function. As a consequence, comparatively to the relational semantics, where there is no edge relation, some vertices are missing. Recovering these vertices is essential for the purpose of reconstructing proofs/terms from their interpretations. It shall also be useful for the comparison with other semantics, like game semantics. In [BE01], Bucciarelli and Ehrhard introduced a so called non uniform coherence space semantics where no vertex is missing. By constructing the co-free exponential we set a new version of this last semantics, together with non uniform versions of hypercoherences and multicoherences, a new semantics where an edge is a finite multiset. Thanks to the co-free construction, these non uniform semantics are deterministic in the sense that the intersection of a clique and of an anti-clique contains at most one vertex, a result of interaction, and extensionally collapse onto the corresponding uniform semantics.Comment: 32 page

Sequential algorithms and strongly stable functions

International audienceIntuitionistic proofs (or PCF programs) may be interpreted as functions between domains, or as strategies on games. The two kinds of interpretation are inherently different: static vs. dynamic, extensional vs. intentional. It is extremely instructive to compare and to connect them. In this article, we investigate the extensional content of the sequential algorithm hierarchy [-] introduced by Berry and Curien two decades ago. We equip every sequential game [T] of the hierarchy with a realizability relation between plays and extensions. In this way, the sequential game [T] becomes a directed acyclic graph, instead of a tree. This enables to define a hypergraph [[T]] on the extensions (or terminal leaves) of the game [T]. We establish that the resulting hierarchy [[T]] coincides with the strongly stable hierarchy introduced by Bucciarelli and Ehrhard. We deduce from this a game-theoretic proof of Ehrhard's collapse theorem, which states that the strongly stable hierarchy coincides with the extensional collapse of the sequential algorithm hierarchy

Nominal Models of Linear Logic

PhD thesisMore than 30 years after the discovery of linear logic, a simple fully-complete model has still not been established. As of today, models of logics with type variables rely on di-natural transformations, with the intuition that a proof should behave uniformly at variable types. Consequently, the interpretations of the proofs are not concrete. The main goal of this thesis was to shift from a 2-categorical setting to a first-order category. We model each literal by a pool of resources of a certain type, that we encode thanks to sorted names. Based on this, we revisit a range of categorical constructions, leading to nominal relational models of linear logic. As these fail to prove fully-complete, we revisit the fully-complete game-model of linear logic established by Melliès. We give a nominal account of concurrent game semantics, with an emphasis on names as resources. Based on them, we present fully complete models of multiplicative additive tensorial, and then linear logics. This model extends the previous result by adding atomic variables, although names do not play a crucial role in this result. On the other hand, it provides a nominal structure that allows for a nominal relationship between the Böhm trees of the linear lambda-terms and the plays of the strategies. However, this full-completeness result for linear logic rests on a quotient. Therefore, in the final chapter, we revisit the concurrent operators model which was first developed by Abramsky and Melliès. In our new model, the axiomatic structure is encoded through nominal techniques and strengthened in such a way that full completeness still holds for MLL. Our model does not depend on any 2-categorical argument or quotient. Furthermore, we show that once enriched with a hypercoherent structure, we get a static fully complete model of MALL

La Ludique : une théorie de l'interaction, de la logique mathématique au langage naturel

Le contenu de ce texte est organisé autour de trois chapitres. Dans le premier, introductif, nous présentons les objets principaux de la Ludique ainsi qu’ils ont été introduits par J.-Y. Girard dans l’article séminal. Les desseins ont été obtenus au terme d’une déconstruction et d’une abstraction de l’objet preuve commesupport de l’interaction. Nous rappelons les principaux résultats constituant l’ossature de la théorie. Nous présentons enfin la reconstruction de la logique traditionnelle dans le cadreludique. Dans ce chapitre, le seul résultat original présenté est un travail effectué en collaboration avec M.-R. Fleury visant à étendre lerésultat de complétude aux formules d’un calcul des prédicats de la Logique Linéaire additive, multiplicative du second ordre.Le second chapitre s’intitule Ludique et théorie du calcul. Nous présentons dans ce chapitreles résultats obtenus et les pistes actuellement abordées autour de l’exploration dela théorie ludique elle-même. En collaboration avec Christophe Fouqueré, nous noussommes attachés à étudier le concept d’incarnation. Nous avons montré dans comment le calcul de l’incarnation du comportement engendré par un ensemble dedesseins était possible sans qu’il soit nécessaire de calculer ce comportement. Nouspoursuivons actuellement notre exploration de la Ludique en vue de comprendrequelles sont dans ce cadre les frontières entre ce qui relève de la Logique linéaire(multiplicative additive) et ce qui n’en relèverait pas. Peut-on caractériser, parmiles comportements, ceux qui sont décomposables selon la grammaire des formuleslinéaires ? Et alors, peut-on caractériser d’autres décompositions et retrouver desconstructions pertinentes dans le cadre de la théorie des types ?Nous nous attachons, dans le troisième chapitre : Ludique et langage naturela mettre en évidence une autre potentialité de la Ludique : sa pertinence pour constituerun cadre théorique propre à la formalisation de différents aspects des languesnaturelles. La Ludique a été utilisée dans une série d’articles afin de rendre compte de différents rentsaspects du langage naturel : de la sémantique à l’argumentation , en passant par les figures du discours. Nous reconstruisons dans ce chapitrel’exposé de cette formalisation ludique des dialogues en langue naturelle

Parallel and serial hypercoherences

International audienc