On the meaning of focalization

Abstract In this paper, we use Girard's Ludics to analyze focalization, a fundamental property of linear logic. In particular, we show how this can be realized interactively thanks to section-retraction pairs (u αβ , f αβ ) between behaviours α ˆ(β Y ), X and αβ Y, X

Inductive and Functional Types in Ludics

Ludics is a logical framework in which types/formulas are modelled by sets of terms with the same computational behaviour. This paper investigates the representation of inductive data types and functional types in ludics. We study their structure following a game semantics approach. Inductive types are interpreted as least fixed points, and we prove an internal completeness result giving an explicit construction for such fixed points. The interactive properties of the ludics interpretation of inductive and functional types are then studied. In particular, we identify which higher-order functions types fail to satisfy type safety, and we give a computational explanation

Incompatibility Semantics from Agreement

In this paper, I discuss the analysis of logic in the pragmatic approach recently proposed by Brandom. I consider different consequence relations, formalized by classical, intuitionistic and linear logic, and I will argue that the formal theory developed by Brandom, even if provides powerful foundational insights on the relationship between logic and discursive practices, cannot account for important reasoning patterns represented by non-monotonic or resource-sensitive inferences. Then, I will present an incompatibility semantics in the framework of linear logic which allow to refine Brandom’s concept of defeasible inference and to account for those non-monotonic and relevant inferences that are expressible in linear logic. Moreover, I will suggest an interpretation of discursive practices based on an abstract notion of agreement on what counts as a reason which is deeply connected with linear logic semantics

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

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

Ludics with repetitions (Exponentials, Interactive types and Completeness)

Ludics is peculiar in the panorama of game semantics: we first have thedefinition of interaction-composition and then we have semantical types, as aset of strategies which "behave well" and react in the same way to a set oftests. The semantical types which are interpretations of logical formulas enjoya fundamental property, called internal completeness, which characterizesludics and sets it apart also from realizability. Internal completeness entailsstandard full completeness as a consequence. A growing body of work start toexplore the potential of this specific interactive approach. However, ludicshas some limitations, which are consequence of the fact that in the originalformulation, strategies are abstractions of MALL proofs. On one side, norepetitions are allowed. On the other side, the proofs tend to rely on the veryspecific properties of the MALL proof-like strategies, making it difficult totransfer the approach to semantical types into different settings. In thispaper, we provide an extension of ludics which allows repetitions and show thatone can still have interactive types and internal completeness. From this, weobtain full completeness w.r.t. a polarized version of MELL. In our extension,we use less properties than in the original formulation, which we believe is ofindependent interest. We hope this may open the way to applications of ludicsapproach to larger domains and different settings