The Cubical Homology of Trace Monoids

Trace monoids have found many applications in computer science [3], [19]. M. Bednarczyk [2] studied and applied the category of asynchronous systems. The author has proved that any asynchronous system can be regarded as a partial trace monoid with action on a set. It allows us to build homology theory for the category of asynchronous systems and Petri nets [9]. It should be noted that the homology theory was introduced and studied for higher dimensional automata in [6]. E. Haucourt [7] applied the Baues-Wirsching homology. The paper is a survey of the author’s results on the homology groups of models for concurrency. We study the relationship between the cubical homology of generalized tori and homology of a trace monoid action on a set. We build the algorithms for computing the homology groups of asynchronous systems, elementary Petri nets, and Mazurkiewicz trace languages. It allows us to solve the problem posed in [9, Open problem 1] constructing an algorithm for computing homology groups of the elementary Petri nets.