Variants of Higher-Dimensional Automata

TL;DR

This paper surveys various weaker variants of higher-dimensional automata (HDAs), analyzing their properties, relationships, and language classes, and demonstrates that partial HDAs can be determinized and admit a Kleene theorem.

Contribution

It consolidates definitions of multiple HDA variants, explores their properties, and establishes key results like language classification, determinization, and Kleene theorem applicability.

Findings

Languages of HDA variants collapse into two classes.

Partial HDAs admit a Kleene theorem.

Partial HDAs are determinizable.

Abstract

The theory of higher-dimensional automata (HDAs) has seen rapid progress in recent years, and first applications, notably to Petri net analysis, are starting to show. It has, however, emerged that HDAs themselves often are too strict a formalism to use and reason about. In order to solve specific problems, weaker variants of HDAs have been introduced, such as HDAs with interfaces, partial HDAs, ST-automata or even relational HDAs. In this paper we collect definitions of these and a few other variants into a coherent whole and explore their properties and translations between them. We show that with regard to languages, the spectrum of variants collapses into two classes, languages closed under subsumption and those that are not. We also show that partial HDAs admit a Kleene theorem and that, contrary to HDAs, they are determinizable.