Sound and complete axiomatizations of coalgebraic language equivalence

Coalgebras provide a uniform framework to study dynamical systems, including several types of automata. In this paper, we make use of the coalgebraic view on systems to investigate, in a uniform way, under which conditions calculi that are sound and complete with respect to behavioral equivalence can be extended to a coarser coalgebraic language equivalence, which arises from a generalised powerset construction that determinises coalgebras. We show that soundness and completeness are established by proving that expressions modulo axioms of a calculus form the rational fixpoint of the given type functor. Our main result is that the rational fixpoint of the functor $FT$, where $T$ is a monad describing the branching of the systems (e.g. non-determinism, weights, probability etc.), has as a quotient the rational fixpoint of the "determinised" type functor $\bar F$, a lifting of $F$ to the category of $T$-algebras. We apply our framework to the concrete example of weighted automata, for which we present a new sound and complete calculus for weighted language equivalence. As a special case, we obtain non-deterministic automata, where we recover Rabinovich's sound and complete calculus for language equivalence.Comment: Corrected version of published journal articl

Divergence-Preserving Branching Bisimilarity

This note considers the notion of divergence-preserving branching bisimilarity. It briefly surveys results pertaining to the notion that have been obtained in the past one-and-a-half decade, discusses its role in the study of expressiveness of process calculi, and concludes with some suggestions for future work.Comment: In Proceedings EXPRESS/SOS 2020, arXiv:2008.1241

The Proof Technique of Unique Solutions of Contractions

International audienceWe review some recent work aimed at understanding proof techniques for behavioural equivalence on processes based on the concept of unique solution of equations. The schema of equations is refined to that of contraction, based on partial orders rather than equalities

A specification language for Reo connectors

Recent approaches to component-based software engineering employ coordinating connectors to compose components into software systems. Reo is a model of component coordination, wherein complex connectors are constructed by composing various type

On the axiomatizability of impossible futures

A general method is established to derive a ground-complete axiomatization for a weak semantics from such an axiomatization for its concrete counterpart, in the context of the process algebra BCCS. This transformation moreover preserves omega-completeness. It is applicable to semantics at least as coarse as impossible futures semantics. As an application, ground- and omega-complete axiomatizations are derived for weak failures, completed trace and trace semantics. We then present a finite, sound, ground-complete axiomatization for the concrete impossible futures preorder, which implies a finite, sound, ground-complete axiomatization for the weak impossible futures preorder. In contrast, we prove that no finite, sound axiomatization for BCCS modulo concrete and weak impossible futures equivalence is ground-complete. If the alphabet of actions is infinite, then the aforementioned ground-complete axiomatizations are shown to be omega-complete. If the alphabet is finite, we prove that the inequational theories of BCCS modulo the concrete and weak impossible futures preorder lack such a finite basis

Equations, Contractions, and Unique Solutions

International audienceOne of the most studied behavioural equivalences is bisimilarity. Its success is much due to the associated bisimulation proof method, which can be further enhanced by means of 'bisimulation up-to' techniques such as 'up-to context'. A different proof method is discussed, based on unique solution of special forms of inequations called contractions, and inspired by Milner's theorem on unique solution of equations. The method is as powerful as the bisimulation proof method and its 'up-to context' enhancements. The definition of contraction can be transferred onto other behavioural equivalences , possibly contextual and non-coinductive. This enables a coinduc-tive reasoning style on such equivalences, either by applying the method based on unique solution of contractions, or by injecting appropriate contraction preorders into the bisimulation game. The techniques are illustrated on CCS-like languages; an example dealing with higher-order languages is also shown

A Complete Axiomatization of Simulation for Regular CCS Expressions

This paper gives axiomatizations of strong and weak simulation over regular CCS expressions. The proof of completeness of the axiomatization of strong simulation is inspired by Milner's proof of completeness of his axiomatization for strong equivalence over regular CCS expressions. Soundness and completeness of the axiomatization for weak simulation is easily deduced from the corresponding result for the axiomatization of strong simulation over regular CCS expressions