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Homological properties of non-deterministic branchings of mergings in higher dimensional automata. (English) Zbl 1085.55003
Higher dimensional automata model concurrent processes. These in turn can be modelled by the category of flows. In this category there exists a notion of “direction” corresponding to that of the flow. There is an analogy to homotopy called dihomotopy which is intended to be such that two dihomotopically equivalent flows will have the same computer-scientific properties. The goal of this paper is to define two dihomotopy invariants $$H_{*}^{-}(X)$$ (resp. $$H_{*}^{+}(X)$$) called the branching (resp. merging) homology of the flow.
The main result of the paper is to show for these homology theories the existence of a long exact sequence associated to a morphism of flows. The method relies on showing that associated to a flow on a space $$X$$ there are topological spaces $${\mathbb P}^{-}X$$ (resp. $${\mathbb P}^{+}X)$$ unique up to homeomorphism that represent the branching (resp. merging) flows faithfully enough that their homology gives the desired homology theory.
The paper is mostly self-contained although there are a couple of places where uses of notation precede the definition. The paper includes some illuminating examples at the end.

##### MSC:
 55N35 Other homology theories in algebraic topology 55P99 Homotopy theory 68Q70 Algebraic theory of languages and automata 68Q85 Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.)
##### Keywords:
flow; automata; merging homology; dihomotopy; branching homology
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