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The notion of innocent strategy was introduced by Hyland and Ong in order to capture the interactive behaviour of λ-terms and PCF programs. An innocent strategy is defined as an alternating strategy with partial memory, in which the strategy plays according to its view. Extending the definition to nonalternating strategies is problematic, because the traditional definition of views is based on the hypothesis that Opponent and Proponent alternate during the interaction. Here, we take advantage of the diagrammatic reformulation of alternating innocence in asynchronous games, in order to provide a tentative definition of innocence in non-alternating games. The task is interesting, and far from easy. It requires the combination of true concurrency and game semantics in a clean and organic way, clarifying the relationship between asynchronous games and concurrent games in the sense of Abramsky and Melliès. It also requires an interactive reformulation of the usual acyclicity criterion of linear logic, as well as a directed variant, as a scheduling criterion. IntroductionThe alternating origins of game semantics. Game semantics was invented (or reinvented) at the beginning of the 1990s in order to describe the dynamics of proofs and programs. It proceeds according to the principles of trace semantics in concurrency theory: every program and proof is interpreted by the sequences of interactions, called plays, that it can have with its environment. The novelty of game semantics is that this set of plays defines a strategy which reflects the interactive behaviour of the program inside the game specified by the type of the program.Game semantics was originally influenced by a pioneering work by Joyal [16] building a category of games (called Conway games) and alternating strategies. In this setting, a game is defined as a decision tree (or more precisely, a dag) in which every edge, called move, has a polarity indicating whether it is played by the program, called Proponent, or by the environment, called Opponent. A play is alternating when Proponent and Opponent alternate strictly -that is, when neither of them plays two moves in a row. A strategy is alternating when it contains only alternating plays.The category of alternating strategies introduced by Joyal was later refined by Abramsky and Jagadeesan [2] in order to characterize the dynamic behaviour of proofs in (multiplicative) linear logic. The key idea is that the tensor product of linear logic, noted ⊗, may be distinguished from its dual, noted Γ, by enforcing a switching policy on plays -ensuring for instance that a strategy of A ⊗ B reacts to an Opponent move played in the subgame A by playing a Proponent move in the same subgame A. ⋆ This work has been supported by the ANR Invariants algébriques des systèmes informatiques (INVAL). Physical address: Équipe PPS, CNRS and Université Paris 7, 2 place Jussieu, case 7017, 75251 Paris cedex 05, France.

We introduce a dependent type theory whose models are weak ω-categories, generalizing Brunerie's definition of ω-groupoids. Our type theory is based on the definition of ω-categories given by Maltsiniotis, himself inspired by Grothendieck's approach to the definition of ω-groupoids. In this setup, ω-categories are defined as presheaves preserving globular colimits over a certain category, called a coherator. The coherator encodes all operations required to be present in an ω-category: both the compositions of pasting schemes as well as their coherences. Our main contribution is to provide a canonical type-theoretical characterization of pasting schemes as contexts which can be derived from inference rules. Finally, we present an implementation of a corresponding proof system.

Abstract. String rewriting systems have proved very useful to study monoids. In good cases, they give finite presentations of monoids, allowing computations on those and their manipulation by a computer. Even better, when the presentation is confluent and terminating, they provide one with a notion of canonical representative of the elements of the presented monoid. Polygraphs are a higher-dimensional generalization of this notion of presentation, from the setting of monoids to the much more general setting of n-categories. One of the main purposes of this article is to give a progressive introduction to the notion of higher-dimensional rewriting system provided by polygraphs, and describe its links with classical rewriting theory, string and term rewriting systems in particular. After introducing the general setting, we will be interested in proving local confluence for polygraphs presenting 2-categories and introduce a framework in which a finite 3-dimensional rewriting system admits a finite number of critical pairs.Recent developments in category theory have established higher-dimensional categories as a fundamental theoretical setting in order to study situations arising in various areas of mathematics, physics and computer science. A nice survey of these can be found in [2], explaining how the use of category theory enables one to unify these apparently unrelated fields of science, by revealing that their intrinsic algebraic structures are in fact closely connected. In the last decade, higher-dimensional categories have therefore emerged as a tool of everyday use for many scientists. The motivation behind the concept of higher dimensions here is that, in order to have a fine-grained understanding of the algebraic structures at stake, one should not only consider morphisms between objects involved, but also morphisms between morphisms (i.e. 2-dimensional morphisms), morphisms between morphisms between morphisms (i.e. 3-dimensional morphisms), and so on. For example, the starting point of algebraic topology [16] is that one should not consider points and paths between them in topological spaces, but also homotopies between paths, and can be refined by also considering homotopies between homotopies and so on.The categorical structures considered nowadays are thus becoming more and more complex, which enables them to capture many details, but the proofs are becoming more and more complicated too, and we are facing the urge for new tools, both of a theoretical and 2012 ACM CCS: [Theory of computation]: Formal languages and automata theory-FormalismsRewrite systems.

State-space reduction techniques, used primarily in model-checkers, all rely on the idea that some actions are independent, hence could be taken in any (respective) order while put in parallel, without changing the semantics. It is thus not necessary to consider all execution paths in the interleaving semantics of a concurrent program, but rather some equivalence classes. The purpose of this paper is to describe a new algorithm to compute such equivalence classes, and a representative per class, which is based on ideas originating in algebraic topology. We introduce a geometric semantics of concurrent languages, where programs are interpreted as directed topological spaces, and study its properties in order to devise an algorithm for computing dihomotopy classes of execution paths. In particular, our algorithm is able to compute a control-flow graph for concurrent programs, possibly containing loops, which is "as reduced as possible" in the sense that it generates traces modulo equivalence. A preliminary implementation was achieved, showing promising results towards efficient methods to analyze concurrent programs, with very promising results compared to partial-order reduction techniques.

International audienceA wide variety of models for concurrent programs has been proposed during the past decades, each one focusing on various aspects of computations: trace equivalence, causality between events, conflicts and schedules due to resource accesses, etc. More recently, models with a geometrical flavor have been introduced, based on the notion of cubical set. These models are very rich and expressive since they can represent commutation between any bunch of events, thus generalizing the principle of true concurrency. While they seem to be very promising - because they make possible the use of techniques from algebraic topology in order to study concurrent computations - they have not yet been precisely related to the previous models, and the purpose of this paper is to fill this gap. In particular, we describe an adjunction between Petri nets and cubical sets which extends the previously known adjunction between Petri nets and asynchronous transition systems by Nielsen and Winskel

International audienceHybrid systems are a widely used model to represent and reason about control-command systems. In an industrial context, these are often implemented in Simulink and their validity is checked by performing many numerical simulations in order to test their behavior with various possible inputs. In this article, we present a tool named HySon which performs set-based simulation of hybrid systems with uncertain parameters, expressed in Simulink. Our tool handles advanced features such as non-linear operations, zero-crossing events or discrete sampling. It is based on well-known efficient numerical algorithms that were adapted to handle set-based domains. We demonstrate the performance of our method on various examples

We study the dependent type theory CaTT, introduced by Finster and Mimram, which presents the theory of weak ω-categories, following the idea that type theories can be considered as presentations of generalized algebraic theories. Our main contribution is a formal proof that the models of this type theory correspond precisely to weak ω-categories, as defined by Maltsiniotis, by generalizing a definition proposed by Grothendieck for weak ωgroupoids: Those are defined as suitable presheaves over a cat-coherator, which is a category encoding structure expected to be found in an ω-category. This comparison is established by proving the initiality conjecture for the type theory CaTT, in a way which suggests the possible generalization to a nerve theorem for a certain class of dependent type theories