Ludics with Repetitions (Exponentials, Interactive Types and Completeness)

Abstract

Abstract. Ludics is peculiar in the panorama of game semantics: we first have the definition of interaction-composition and then we have semantical types, as a set of strategies which "behave well" and react in the same way to a set of tests. The semantical types which are interpretations of logical formulas enjoy a fundamental property, called internal completeness, which characterizes ludics and sets it apart also from realizability. Internal completeness entails standard full completeness as a consequence.A growing body of work start to explore the potential of this specific interactive approach. However, ludics has some limitations, which are consequence of the fact that in the original formulation, strategies are abstractions of MALL proofs. On one side, no repetitions are allowed. On the other side, the proofs tend to rely on the very specific properties of the MALL proof-like strategies, making it difficult to transfer the approach to semantical types into different settings.In this paper, we provide an extension of ludics which allows repetitions and show that one can still have interactive types and internal completeness. From this, we obtain 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 of independent interest. We hope this may open the way to applications of ludics approach to larger domains and different settings.

An Infinitary Affine Lambda-Calculus Isomorphic to the Full Lambda-Calculus

“…However, the same ideas presented here may be applied, mutatis mutandis, to Girard's ludics [16], an untyped, gametheoretic framework from which multiplicative additive linear logic may be recovered. Our preliminary results give a new solution to the question of modeling the exponential modalities in ludics, alternative to the one proposed by Basaldella and Faggian [6]. For the reader familiar with the terminology of ludics, our solution is based on infinite ramifications, which are consistent with the fact that, read bottom-up, an exponential rule creates unboundedly many loci for unlimited (affine) use during the rest of the proof.…”

A proof theoretical framework for argumentation modeling

“…Girard est « linéaire », i.e. les lieux ne peuvent être utilisés au plus qu'une fois, les travaux récents pallient cette limitation : la Ludique avec répétitions [Basaldella, Faggian, 2009], l'extension de la Ludique aux desseins non linéaires [Terui, 2011] pourraient nous fournir le cadre théorique pertinent.…”

“…D'une certaine manière, la dé-marche d'une formalisation des dialogues en Ludique initiée par Lecomte et Quatrini [2010], que nous reprenons ici, s'inscrit dans cette perspective. En effet la Ludique est non seulement une théorie logique, mais elle peut être recomposée avec des concepts de la sémantique des jeux [Basaldella, Faggian, 2009]. Toutefois, la Ludique est d'abord une théorie de l'interaction.…”

“…De telles séquences s'appellent en Ludique des chroniques. Suivant la métaphore des jeux, ces séquences alternées d'actions peuvent être vues comme des parties que l'on peut regrouper pour former des stratégies nommées desseins en Ludique [Basaldella, Faggian, 2009]. Les desseins sont ainsi des ensembles de chroniques qui sont susceptibles de s'intercaler pour supporter les étapes successives d'un même parcours d'interaction.…”

The Lambda Lambda-Bar calculus

“…The difference is that single threaded strategies may have more than one initial move; and are "weakly innocent" for each of these. 5 We now use Innocent( φ ) as the basis for the definition of να.φ . On one hand, the strategy Innocent( φ ) reacts to its initial move as many time as asked, by behaving as φ; on the other hand, it contains moves corresponding to the distinguished variable α.…”