Languages of Higher-Dimensional Automata

TL;DR

This paper introduces a formal framework for languages of higher-dimensional automata (HDAs), establishing their properties, structure, and how they relate to interval orders, with implications for automata theory.

Contribution

It defines a new category of precubical sets and introduces event consistency, linking HDAs to interval orders and demonstrating their language properties.

Findings

Languages of HDAs are sets of interval orders closed under subsumption

Any finite subsumption-closed set of interval orders is realizable as an HDA language

Languages of HDAs are closed under union and parallel composition

Abstract

We introduce languages of higher-dimensional automata (HDAs) and develop some of their properties. To this end, we define a new category of precubical sets, uniquely naturally isomorphic to the standard one, and introduce a notion of event consistency. HDAs are then finite, labeled, event-consistent precubical sets with distinguished subsets of initial and accepting cells. Their languages are sets of interval orders closed under subsumption; as a major technical step we expose a bijection between interval orders and a subclass of HDAs. We show that any finite subsumption-closed set of interval orders is the language of an HDA, that languages of HDAs are closed under binary unions and parallel composition, and that bisimilarity implies language equivalence.