A Simplicial Model for KB4nKB4_n: Epistemic Logic with Agents that May Die

The standard semantics of multi-agent epistemic logic S5 is based on Kripke models whose accessibility relations are reflexive, symmetric and transitive. This one dimensional structure contains implicit higher-dimensional information beyond pairwise interactions, that we formalized as pure simplicial models in a previous work (Information and Computation, 2021). Here we extend the theory to encompass simplicial models that are not necessarily pure. The corresponding class of Kripke models are those where the accessibility relation is symmetric and transitive, but might not be reflexive. Such models correspond to the epistemic logic KB4 . Impure simplicial models arise in situations where two possible worlds may not have the same set of agents. We illustrate it with distributed computing examples of synchronous systems where processes may crash

A simplicial model for KB4n : epistemic logic with agents that may die

The standard semantics of multi-agent epistemic logic S5n is based on Kripke models whose accessibility relations are reflexive, symmetric and transitive. This one dimensional structure contains implicit higher-dimensional information beyond pairwise interactions, that we formalized as pure simplicial models in a previous work in Information and Computation 2021 [10]. Here we extend the theory to encompass simplicial models that are not necessarily pure. The corresponding class of Kripke models are those where the accessibility relation is symmetric and transitive, but might not be reflexive. Such models correspond to the epistemic logic KB4n. Impure simplicial models arise in situations where two possible worlds may not have the same set of agents. We illustrate it with distributed computing examples of synchronous systems where processes may crash

Formal Relationships Between Geometrical and Classical Models for Concurrency

A 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

Computing the vertices of tropical polyhedra using directed hypergraphs

We establish a characterization of the vertices of a tropical polyhedron defined as the intersection of finitely many half-spaces. We show that a point is a vertex if, and only if, a directed hypergraph, constructed from the subdifferentials of the active constraints at this point, admits a unique strongly connected component that is maximal with respect to the reachability relation (all the other strongly connected components have access to it). This property can be checked in almost linear-time. This allows us to develop a tropical analogue of the classical double description method, which computes a minimal internal representation (in terms of vertices) of a polyhedron defined externally (by half-spaces or hyperplanes). We provide theoretical worst case complexity bounds and report extensive experimental tests performed using the library TPLib, showing that this method outperforms the other existing approaches.Comment: 29 pages (A4), 10 figures, 1 table; v2: Improved algorithm in section 5 (using directed hypergraphs), detailed appendix; v3: major revision of the article (adding tropical hyperplanes, alternative method by arrangements, etc); v4: minor revisio

Order-Theoretic, Geometric and Combinatorial Models of Intuitionistic S4 Proofs

We propose a few models of proof terms for the intuitionistic modal propositional logic S4. Some of them are based on partial orders, or cpos, or dcpos, some of them on a suitable category of topological spaces and continuous maps. A structure that emerges from these interpretations is that of augmented simplicial sets. This leads to so-called combinatorial models, where simplices play an important role: the point is that the simplicial structure interprets the 2 modality, and that the category of augmented simplicial sets is itself already a model of intuitionistic propositional S4 proof terms. In fact, this category is an elementary topos, and is therefore a prime candidate to interpret all proof terms for intuitionistic S4 set theory. Finally, we suggest that geometric-like realizations functors provide a recipe to build other models of intuitionistic propositional S4 proof terms. 1 Introduction There are now several different proof term languages for intuitionistic S4 [BdP92, BdP9..

Natural Homology

International audienc

Static Analysis of Numerical Programs (Constrained Affine Sets Abstract Domain)

Nous nous plaçons dans le cadre de l'analyse statique de programmes, et nous nous intéressons aux propriétés numériques, c'est a dire celles qui concernent les valeurs numériques des variables de programmes. Nous essayons en particulier de déterminer une sur-approximation garantie de l'ensemble de valeurs possibles pour chaque variable numérique utilisée dans le programme à analyser. Cette analyse statique est faite dans le cadre de la théorie de l'interprétation abstraite, théorie présentant un compromis entre les limites théoriques d'indécidabilite et de calculabilite et la précision des résultats obtenus. Nous sommes partis des travaux d'Eric Goubault et Sylvie Putot, que nous avons étendus et généralisés. Notre nouveau domaine abstrait, appelé ensembles affines contraints, combine à la fois l'efficacite de calcul des domaines à base de formes affines et le pouvoir ex- pressif des domaines relationnels classiques tels que les octogones ou les polyèdres. Le nouveau domaine a été implémenté pour mettre en évidence l'intérêt de cette combinaison, ses avantages, ses performances et ses limites par rapport aux autres domaines numériques déjà existants. Le formalisme ainsi que les résultats pra- tiques ont fait l'objet de plusieurs publications [CAV 2009, CAV 2010]We aim at proving automatically the correctness of numerical behavior of a program by inferring invariants on numerical variables. More precisely, we over-approximate in a sound manner the set of reached values. We use Abstract Interpretation-based Static Analysis as a generic framework to de ne and ap- proximate the semantics of a program in a uni ed manner. The semantics that describe the real behavior of the program (concrete semantics) is in general unde- cidable. Abstract interpretation o ers a way to abstract this concrete semantics to obtain a decidable semantics involving machine-expressible objects. We in- troduce a new affine forms-based abstract domain, called constrained affine sets, which extends and generalizes an already existing abstract domain introduced by Eric Goubault and Sylvie Putot. The expressiveness of such new domain is enhanced thanks to its ability to encode and propagate linear constraints among variables. We have implemented our new domain to experiment the precision and the efficiency of our approach and compare our results to the already existing abstract domains. The theoretical work as well as the implementation and the experiments have been the subject of two publications [CAV 2009, CAV 2010]PALAISEAU-Polytechnique (914772301) / SudocSudocFranceF