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The Freeness Problem for Automaton Semigroups

Daniele D'Angeli, Emanuele Rodaro, Jan Philipp Wächter

TL;DR

This work proves that the freeness problem for automaton semigroups and automaton monoids is undecidable, solving an open question by encoding Post's Correspondence Problem (PCP) into automaton-generated semigroups/monoids. A novel PCP-to-automaton encoding is developed that works even with a fixed alphabet size, enabling precise control over relations in the generated algebra and yielding broader undecidability results such as left cancellativity and equidivisibility failing, as well as undecidability for extending homomorphisms. The results extend to the free presentation problem for automaton monoids and imply undecidability of related decision problems, with a journal-version detailing the full constructions. The methodology demonstrates the versatility of automaton-based encodings and hints at future avenues toward undecidability in automaton groups, while leaving open whether the alphabet size can be reduced further (e.g., to binary) for these undecidability results. Overall, the paper significantly deepens our understanding of the algorithmic limits in automaton-generated (semi)groups and monoids and provides a robust framework for deriving further undecidability results in related algebraic structures.

Abstract

We show that the freeness problems for automaton semigroups and for automaton monoids are undecidable and, thereby, solve an open problem listed by Grigorchuk, Nekrashevych and Sush\-chanskĭi. We achieve this using a new technique to encode Post's Correspondence Problem into automaton semigroups and monoids and our result even holds if we restrict the alphabet of the input automata to a constant size. The encoding allows us to precisely control the relations in the generated semigroup/monoid and the construction is quite versatile. In fact, we obtain further undecidability results on various semigroup notions (left cancellativity, equidivisibility and extending homomorphisms). Our construction can also be adapted to show that the free presentation problem for automaton monoids is undecidable (and yields a weaker statement in the semigroup case).

The Freeness Problem for Automaton Semigroups

TL;DR

This work proves that the freeness problem for automaton semigroups and automaton monoids is undecidable, solving an open question by encoding Post's Correspondence Problem (PCP) into automaton-generated semigroups/monoids. A novel PCP-to-automaton encoding is developed that works even with a fixed alphabet size, enabling precise control over relations in the generated algebra and yielding broader undecidability results such as left cancellativity and equidivisibility failing, as well as undecidability for extending homomorphisms. The results extend to the free presentation problem for automaton monoids and imply undecidability of related decision problems, with a journal-version detailing the full constructions. The methodology demonstrates the versatility of automaton-based encodings and hints at future avenues toward undecidability in automaton groups, while leaving open whether the alphabet size can be reduced further (e.g., to binary) for these undecidability results. Overall, the paper significantly deepens our understanding of the algorithmic limits in automaton-generated (semi)groups and monoids and provides a robust framework for deriving further undecidability results in related algebraic structures.

Abstract

We show that the freeness problems for automaton semigroups and for automaton monoids are undecidable and, thereby, solve an open problem listed by Grigorchuk, Nekrashevych and Sush\-chanskĭi. We achieve this using a new technique to encode Post's Correspondence Problem into automaton semigroups and monoids and our result even holds if we restrict the alphabet of the input automata to a constant size. The encoding allows us to precisely control the relations in the generated semigroup/monoid and the construction is quite versatile. In fact, we obtain further undecidability results on various semigroup notions (left cancellativity, equidivisibility and extending homomorphisms). Our construction can also be adapted to show that the free presentation problem for automaton monoids is undecidable (and yields a weaker statement in the semigroup case).
Paper Structure (19 sections, 23 theorems, 31 equations, 11 figures, 2 tables)

This paper contains 19 sections, 23 theorems, 31 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properties of Free Semigroups and Monoids.
  4. The Freeness Problem for Automaton Semigroups and Monoids
  5. Definition of the Automaton $\hat{\mathcal{R}}$.
  6. Definition of the Automaton $\mathcal{S}$.
  7. Definition of the Automaton $\mathcal{T}$.
  8. Correctness.
  9. Converse Direction.
  10. Main Theorem and other Consequences.
  11. Free Presentations of Monoids
  12. Definition of the Automaton $\hat{\mathcal{R}}$.
  13. Definition of the Automaton $\mathcal{S}$.
  14. Definition of the Automaton $\mathcal{T}$.
  15. Correctness.
  16. ...and 4 more sections

Key Result

Proposition 2.8

On input of a finite set $R$ with $|R| \geq 2$, one may compute a complete $\mathscr{S}$-au-to-ma-ton $\mathcal{R} = (R, \{ 0, 1 \}, \rho)$ (i. e. one with binary alphabet) with $\mathscr{S}(\mathcal{R}) = R^+$ and $\mathscr{M}(\mathcal{R}) = R^*$.

Figures (11)

  • Figure 1: Graphical representation of equidivisibility waechter2020automaton.
  • Figure 2: Example of depicting a transition in an automaton.
  • Figure 3: Combined and abbreviated cross diagrams.
  • Figure 4: A complete $\mathscr{S}$-au-to-ma-ton generating $\{ a, b \}^+$.
  • Figure 5: Automata over binary alphabet generating free (semi)groups.
  • ...and 6 more figures

Theorems & Definitions (59)

  • Remark
  • Remark
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Example 2.6: The Adding Machine
  • Example 2.7
  • Proposition 2.8
  • proof
  • Proposition 2.9
  • ...and 49 more