An Automaton Group with PSPACE-Complete Word Problem

Authors Jan Philipp Wächter , Armin Weiß

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Jan Philipp Wächter
  • Universität Stuttgart, Institut für Formale Methoden der Informatik (FMI), Universitätsstraße 38, 70569 Stuttgart, Germany
Armin Weiß
  • Universität Stuttgart, Institut für Formale Methoden der Informatik (FMI), Universitätsstraße 38, 70569 Stuttgart, Germany

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Jan Philipp Wächter and Armin Weiß. An Automaton Group with PSPACE-Complete Word Problem. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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. 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 circuits of bounded degree and logarithmic depth in the group of even permutations over five elements.

Subject Classification

ACM Subject Classification
  • Theory of computation → Transducers
  • automaton group
  • word problem
  • compressed word problem


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