# An Automaton Group with PSPACE-Complete Word Problem

## TL;DR

This paper constructs an automaton group with a PSPACE-complete word problem, demonstrating the complexity of decision problems in automaton groups and establishing optimality with respect to alphabet size.

## Contribution

It introduces a new automaton group that simulates Turing machine computations, proving PSPACE-completeness of the word problem and EXPSPACE-completeness of the compressed word problem.

## Key findings

- Automaton group with PSPACE-complete word problem
- Constructed group has EXPSPACE-complete compressed word problem
- Group acts over a binary alphabet, showing optimal alphabet size

## Abstract

We construct an automaton group with a PSPACE-complete word problem, proving a conjecture due to Steinberg. Additionally, the constructed group has a provably more difficult, namely EXPSPACE-complete, compressed word problem and acts over a binary alphabet. Thus, it is optimal in terms of the alphabet size. Our construction directly simulates the computation of a Turing machine in an automaton group and, therefore, seems to be quite versatile. It combines two ideas: the first one is a construction used by D'Angeli, Rodaro and the first author to obtain an inverse automaton semigroup with a PSPACE-complete word problem and the second one is to utilize a construction used by Barrington to simulate Boolean circuits of bounded degree and logarithmic depth in the group of even permutations over five elements.