Weak equivalence of higher-dimensional automata

TL;DR

This paper proposes a new notion of weak equivalence for higher-dimensional automata that preserves trace and homology languages, enabling simplification through merging and collapsing while maintaining behavioral properties.

Contribution

It introduces weak equivalence for HDAs, linking trace and homology languages, and demonstrates reduction techniques compatible with automata operations.

Findings

Weak equivalence preserves trace and homology languages.

HDAs can be reduced to smaller equivalent automata.

Weak equivalence is compatible with tensor and coproduct operations.

Abstract

This paper introduces a notion of equivalence for higher-dimensional automata, called weak equivalence. Weak equivalence focuses mainly on a traditional trace language and a new homology language, which captures the overall independence structure of an HDA. It is shown that weak equivalence is compatible with both the tensor product and the coproduct of HDAs and that, under certain conditions, HDAs may be reduced to weakly equivalent smaller ones by merging and collapsing cubes.