Formal Relationships Between Geometrical and Classical Models for Concurrency

TL;DR

This paper establishes a formal connection between geometrical models of concurrency based on cubical sets and classical models like Petri nets, enabling the application of algebraic topology techniques to analyze concurrent computations.

Contribution

It introduces an adjunction between Petri nets and cubical sets, bridging geometric and classical models of concurrency for the first time.

Findings

An adjunction between Petri nets and cubical sets is constructed.

This adjunction extends previous relationships between Petri nets and asynchronous transition systems.

The work enables the use of algebraic topology in studying concurrent computations.

Abstract

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.