On the Complexity of the Word Problem for Automaton Semigroups and Automaton Groups
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.
Keywords
Cite
@article{arxiv.1611.09541,
title = {On the Complexity of the Word Problem for Automaton Semigroups and Automaton Groups},
author = {Daniele D'Angeli and Emanuele Rodaro and Jan Philipp Wächter},
journal= {arXiv preprint arXiv:1611.09541},
year = {2017}
}