Combinatorics of branchings in higher dimensional automata

TL;DR

This paper investigates the combinatorial properties of branching in higher dimensional automata through new homology theories, providing calculations and invariance results to deepen understanding of automata execution path structures.

Contribution

It introduces the reduced branching homology theory, explores its properties, and applies it to compute homology of specific $mbda$-categories, advancing the mathematical understanding of automata branching.

Findings

Calculated branching homology of certain $mbda$-categories.

Proved invariance properties of the reduced branching homology.

Conjectured equivalence between reduced and non-reduced theories for higher automata.

Abstract

We explore the combinatorial properties of the branching areas of execution paths in higher dimensional automata. Mathematically, this means that we investigate the combinatorics of the negative corner (or branching) homology of a globular $\omega$-category and the combinatorics of a new homology theory called the reduced branching homology. The latter is the homology of the quotient of the branching complex by the sub-complex generated by its thin elements. Conjecturally it coincides with the non reduced theory for higher dimensional automata, that is $\omega$-categories freely generated by precubical sets. As application, we calculate the branching homology of some $\omega$-categories and we give some invariance results for the reduced branching homology. We only treat the branching side. The merging side, that is the case of merging areas of execution paths is similar and can be easily deduced from the branching side.