Kleene Theorem for Higher-Dimensional Automata

TL;DR

This paper establishes a Kleene theorem for higher-dimensional automata, characterizing recognized languages as rational, subsumption-closed sets of finite interval pomsets, and introduces new algebraic-topological tools for their analysis.

Contribution

It proves a Kleene theorem for higher-dimensional automata and develops algebraic-topological methods to analyze their recognized languages.

Findings

Languages recognized are rational subsumption-closed sets of finite interval pomsets

Introduces higher-dimensional automata with interfaces modeled as presheaves

Develops tools inspired by algebraic topology, such as cylinders and (co)fibrations

Abstract

We prove a Kleene theorem for higher-dimensional automata. It states that the languages they recognise are precisely the rational subsumption-closed sets of finite interval pomsets. The rational operations on these languages include a gluing composition, for which we equip pomsets with interfaces. For our proof, we introduce higher-dimensional automata with interfaces, which are modelled as presheaves over labelled precube categories, and develop tools and techniques inspired by algebraic topology, such as cylinders and (co)fibrations. Higher-dimensional automata form a general model of non-interleaving concurrency, which subsumes many other approaches. Interval orders are used as models for concurrent and distributed systems where events extend in time. Our tools and techniques may therefore yield templates for Kleene theorems in various models and applications.