The Word Problem for Finitary Automaton Groups

TL;DR

This paper investigates the computational complexity of the word problem in finitary automaton groups, establishing coNP-completeness for the uniform case and PSpace-completeness for the compressed case, with implications for automata over binary alphabets.

Contribution

It proves the complexity classifications of the word problem for finitary automaton groups, including the uniform and compressed variants, using reductions from boolean satisfiability.

Findings

Uniform word problem is coNP-complete.

Compressed word problem is PSpace-complete.

Complexity remains the same for binary alphabet automata.

Abstract

A finitary automaton group is a group generated by an invertible, deterministic finite-state letter-to-letter transducer whose only cycles are self-loops at an identity state. We show that, for this presentation of finite groups, the uniform word problem is coNP-complete. Here, the input consists of a finitary automaton together with a finite state sequence and the question is whether the sequence acts trivially on all input words. Additionally, we also show that the respective compressed word problem, where the state sequence is given as a straight-line program, is PSpace-complete. In both cases, we give a direct reduction from the satisfiability problem for (quantified) boolean formulae and we further show that the problems remain complete for their respective classes if we restrict the input alphabet of the automata to a binary one.