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The Freeness Problem for Automaton Semigroups

Authors Daniele D'Angeli , Emanuele Rodaro , Jan Philipp Wächter

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Author Details

Daniele D'Angeli
  • Università degli Studi Niccolò Cusano, Roma, Italy
Emanuele Rodaro
  • Department of Mathematics, Politecnico di Milano, Italy
Jan Philipp Wächter
  • Fachrichtung Mathematik, Universität des Saarlandes, Saarbrücken, Germany

Funding

  • Wächter, Jan Philipp: funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 492814705 – while visiting the Department of Mathematics at Politecnico di Milano. The listed affiliation is his current one, partly funded by ERC grant 101097307.

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Daniele D'Angeli, Emanuele Rodaro, and Jan Philipp Wächter. The Freeness Problem for Automaton Semigroups. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 44:1-44:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.44

Abstract

We present a new technique to encode Post’s Correspondence Problem into automaton semigroups and monoids. The encoding allows us to precisely control whether there exists a relation in the generated semigroup/monoid and thus show that the freeness problems for automaton semigroups and for automaton monoids (listed as open problems by Grigorchuk, Nekrashevych and Sushchanskĭi) are undecidable. The construction seems to be quite versatile and we obtain the undecidability of further problems: Is a given automaton semigroup (monoid) (left) cancellative? Is it equidivisible (which - together with the existence of a (proper) length function - characterizes free semigroups and monoids)? Does a given map extend into a homomorphism between given automaton semigroups? Finally, our construction can be adapted to show that it is undecidable whether a given automaton generates a free monoid whose basis is given by the states (but where we allow one state to act as the identity). In the semigroup case, we show a weaker version of this.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Automaton Monoid
  • Automaton Semigroup
  • Freeness Problem
  • Free Presentation

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