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On the homology of small categories and asynchronous transition systems. (English) Zbl 1078.18005
The main subject of this paper is devoted to the calculation of homology groups of concurrent computing models. In a previous work, the author, together with V. V. Tkachenko [KhGPU, 23–33 (2003); http://www.Knastu.ru], proved that the category of asyncronous transition systems admits a functor into the category of pointed sets over partially commutative monoids, permitting now to define homology groups for such systems. The interpretation and computation of the first homology group of the small category given by a rewriting system shows that its elements can be viewed as the equivalence classes of the flows in the graph of the rewriting system. Then, the homology groups of asyncronous transition systems and Petri nets are obtained.

MSC:
18B20 Categories of machines, automata
18G10 Resolutions; derived functors (category-theoretic aspects)
68M14 Distributed systems
55U35 Abstract and axiomatic homotopy theory in algebraic topology
68Q85 Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.)
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