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 number of events, thus generalizing the principle of true concurrency. While they are emerging as a central tool in concurrency, which is 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. A great variety of models for concurrency was introduced in the last decades: transition systems (with independence), asynchronous automata, event structures, Petri nets, etc. Each of these models focuses on modeling a particular aspect of computations, and even though their nature are very different, they are tightly related to each other as witnessed in [43]. More recently, models inspired by ideas coming from geometry, such as cubical sets (also sometimes called higher dimensional automata or HDA [30, 18]) or local po-spaces [12], have emerged as central tools to study concurrency: thanks to their nice algebraic structure, they allow one to carry on abstractly many computations, and they are very expressive because of their ability to represent commutations between multiple events. However, since their introduction, they have not been systematically and formally linked with the other models, such as transition systems, even though cubical sets contain a notion of generalized transition in their very definition. ∗CEA, LIST, Point Courrier 94, 91191 Gif-sur-Yvette, France. This work has been supported by the PANDA (“Parallel and Distributed Analysis”, ANR-09-BLAN-0169) French ANR project.