Cubical Homology of Asynchronous Transition Systems

TL;DR

This paper establishes an isomorphism between the homology groups of certain algebraic structures and semicubical sets, providing a finite complex for computing homology of asynchronous transition systems with bounded independence.

Contribution

It introduces a method to compute homology groups of asynchronous transition systems using finite complexes, linking algebraic and topological perspectives.

Findings

Homology groups of locally finite-dimensional free partially commutative monoids are isomorphic to those of semicubical sets.

A finite-length complex is constructed for systems with finite maximal independent events.

Examples demonstrate the practical computation of these homology groups.

Abstract

We show that a set with an action of a locally finite-dimensional free partially commutative monoid and the corresponding semicubical set have isomorpic homology groups. We build a complex of finite length for the computing homology groups of any asynchronous transition system with finite maximal number of mutually independent events. We give examples of computing the homology groups.