On the Complexity of the Word Problem for Automaton Semigroups and Automaton Groups

TL;DR

This paper investigates the computational complexity of the word problem in automaton semigroups, automaton groups, and related structures, establishing PSPACE-completeness results and exploring bounds on distinguishing words.

Contribution

It introduces automaton-inverse semigroups and demonstrates their PSPACE-complete word problem, providing new insights into the complexity of automaton-generated algebraic structures.

Findings

Automaton-inverse semigroups can have PSPACE-complete word problems.

The word problem for automaton groups with a single rational constraint is PSPACE-complete.

The uniform word problem for automaton groups is NL-hard.

Abstract

In this paper, we study the word problem for automaton semigroups and automaton groups from a complexity point of view. As an intermediate concept between automaton semigroups and automaton groups, we introduce automaton-inverse semigroups, which are generated by partial, yet invertible automata. We show that there is an automaton-inverse semigroup and, thus, an automaton semigroup with a PSPACE-complete word problem. We also show that there is an automaton group for which the word problem with a single rational constraint is PSPACE-complete. Additionally, we provide simpler constructions for the uniform word problems of these classes. For the uniform word problem for automaton groups (without rational constraints), we show NL-hardness. Finally, we investigate a question asked by Cain about a better upper bound for the length of a word on which two distinct elements of an automaton semigroup must act differently.