Multipass automata and group word problems

TL;DR

This paper introduces multipass automata, generalizing pushdown automata, and characterizes the classes of languages they accept, linking these classes to group word problems and their computational complexity.

Contribution

It defines multipass automata, characterizes their language classes, and applies these concepts to analyze groups with specific word problem complexities.

Findings

Deterministic multipass automata accept Boolean closures of deterministic context-free languages.

Nondeterministic multipass automata accept poly-context-free languages.

Application to groups shows correspondence between automata classes and group word problems.

Abstract

We introduce the notion of multipass automata as a generalization of pushdown automata and study the classes of languages accepted by such machines. The class of languages accepted by deterministic multipass automata is exactly the Boolean closure of the class of deterministic context-free languages while the class of languages accepted by nondeterministic multipass automata is exactly the class of poly-context-free languages, that is, languages which are the intersection of finitely many context-free languages. We illustrate the use of these automata by studying groups whose word problems are in the above classes.