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arXiv · 1611.09541

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

Daniele D'Angeli·Emanuele Rodaro·Jan Philipp Wächter
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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.

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Daniele D'Angeli, Emanuele Rodaro, Jan Philipp Wächter. 2016-11-29. On the Complexity of the Word Problem for Automaton Semigroups and Automaton Groups. https://doi.org/10.1016/j.aam.2017.05.008

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