Combinatorics of branchings in higher dimensional automata (Q2724148)

By treating the concurrent execution of \(n\) \(1\)-transitions as an \(n\)-cube it is possible to model higher dimensional automata as pre-cubical sets. The two families of face maps in the resulting chain complexes give rise to homology theories whose non-trivial elements model the branching (respectively merging) areas of the execution paths of the automaton. Unfortunately this model distinguishes between automata whose branching behaviours are topologically the same. It is therefore desirable to refine the category of pre-cubical complexes in such a way that automata whose branching behaviours are equivalent have the same homology. This paper defines a new homology theory called the reduced branching homology in which the previously defined branching complex is extended by the addition of ``thin elements''. The author conjectures that the resulting homology theory does indeed capture the required topology. The material presented involves substantial notational and technical details.