Ludics

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.

Abstract: Preface List of contributors Part I. Tutorials: 1. Category theory for linear logicians R. Blute and Ph. Scott 2. Proof nets and the x-calculus S. Guerrini 3. An overview of linear logic programming D. Miller 4. Linearity and nonlinearity in distributed computation G. Winskel Part II. Refereed Articles: 5. An axiomatic approach to structural rules for locative linear logic J. M. Andreoli 6. An introduction to uniformity in ludics C. Faggian, M. R. Fleury-Donnadieu and M. Quatrini 7. Slicing polarized addictive normalization O. Laurent and L. Toratora De Falco 8. A topological correctness criterion for muliplicative noncommutative logic P.A. Mellies Part III. Invited Articles: 9. Bicategories in algebra and linguistics J. Lambek 10. Between logic and quantic: a tract J. Y. Girard.

Abstract: Ludics [1] is a novel approach to logic—especially proof-theory. The present introduction emphasises foundational issues. For ages, not a single disturbing idea in the area of “foundations”: the discussion is sort of ossified—as if everything had been said, as if all notions had taken their definite place, in a big cemetery of ideas. One can still refresh the flowers or regild the stone, e.g., prove technicalities, sometimes non-trivial; but the real debate is still: this paper begins with an autopsy , the autopsy of the foundational project. Up to say 1900, the realist/dualist approach to science was dominant; during the last century some domains like physics evolved so as to become completely anti-realist; but this evolution hardly concerned logic. By the turn of the XX th century mathematics was jeopardised by paradoxes, the most famous of them being due to Russell. Hilbert's reaction was to focus on consistency . But the reduction of paradoxes—and therefore of foundations—to solely the antinomies is highly questionable: indeed, the typical paradoxical artifacts are secret sharers , objects satisfying the formal definitions but far astray from the intended meaning, typically the Peano “curve” which contradicts our perception of dimension . Fortunately, topology has been able to show that dimension m is not the same as dimension n … but just for a second, forget this and imagine consistent mathematics in which balls in any dimension are homeomorphic: what a disaster! This exclusive focus on consistency—not to speak of the strategic failure of the Programme—should explain why logic, especially foundations lost contact with other sciences during last century.

Abstract: We consider the setting of L-nets, recently introduced by Faggian and Maurel as a game model of concurrent interaction and based on Girard's Ludics. We show how L-nets satisfying an additional condition, which we call logical L-nets, can be sequentialized into traditional tree-like strategies, and vice-versa.