Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Finiteness Problem for Automaton Semigroups of Extended Bounded Activity

Daniele D'Angeli, Emanuele Rodaro, Jan Philipp Wächter

TL;DR

The paper extends the activity framework for automaton semigroups to an extended, yet bounded, activity setting, revealing that words with infinite orbits form a deterministic Büchi language. It leverages expandability and a refined expansion relation to construct finite, uniform witnesses (NFRAs) and a deterministic Büchi acceptor, enabling decidability of finiteness questions for complete automaton semigroups/monoids under bounded $S$-activity. The results further permit analysis of subsemigroups defined by regular suffix-closed languages and include dual-automaton consequences such as torsion and finiteness questions, all reducible to $\omega$-regular language emptiness. Collectively, these findings provide a robust algorithmic framework for finiteness problems in a broad class of automaton semigroups beyond groups, using $\omega$-regular language techniques and expandability-based reductions.

Abstract

We extend the notion of activity for automaton semigroups and monoids introduced by Bartholdi, Godin, Klimann and Picantin to a more general setting. Their activity notion was already a generalization of Sidki's activity hierarchy for automaton groups. Using the concept of expandability introduced earlier by the current authors, we show that the language of $ω$-words with infinite orbits is effectively a deterministic Büchi language for our extended activity. This generalizes a similar previous result on automaton groups by Bondarenko and the third author. By a result of Francoeur and the current authors, the description via a Büchi automaton immediately yields that the finiteness problem for complete automaton semigroups and monoids of bounded activity is decidable. In fact, we obtain a stronger result where we may consider sub-orbits under the action of a regular, suffix-closed language over the generators. This, in particular, also yields that it is decidable whether a finitely generated subsemigroup (or -monoid) of a bounded complete automaton semigroup is finite.

The Finiteness Problem for Automaton Semigroups of Extended Bounded Activity

TL;DR

The paper extends the activity framework for automaton semigroups to an extended, yet bounded, activity setting, revealing that words with infinite orbits form a deterministic Büchi language. It leverages expandability and a refined expansion relation to construct finite, uniform witnesses (NFRAs) and a deterministic Büchi acceptor, enabling decidability of finiteness questions for complete automaton semigroups/monoids under bounded -activity. The results further permit analysis of subsemigroups defined by regular suffix-closed languages and include dual-automaton consequences such as torsion and finiteness questions, all reducible to -regular language emptiness. Collectively, these findings provide a robust algorithmic framework for finiteness problems in a broad class of automaton semigroups beyond groups, using -regular language techniques and expandability-based reductions.

Abstract

We extend the notion of activity for automaton semigroups and monoids introduced by Bartholdi, Godin, Klimann and Picantin to a more general setting. Their activity notion was already a generalization of Sidki's activity hierarchy for automaton groups. Using the concept of expandability introduced earlier by the current authors, we show that the language of -words with infinite orbits is effectively a deterministic Büchi language for our extended activity. This generalizes a similar previous result on automaton groups by Bondarenko and the third author. By a result of Francoeur and the current authors, the description via a Büchi automaton immediately yields that the finiteness problem for complete automaton semigroups and monoids of bounded activity is decidable. In fact, we obtain a stronger result where we may consider sub-orbits under the action of a regular, suffix-closed language over the generators. This, in particular, also yields that it is decidable whether a finitely generated subsemigroup (or -monoid) of a bounded complete automaton semigroup is finite.
Paper Structure (13 sections, 10 theorems, 22 equations)

This paper contains 13 sections, 10 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sets, Semigroups, Words and (Regular) Languages.
  4. $\mathscr{S}$-automata.
  5. Orbits.
  6. The Product $\mathcal{T} \circ w \times R$.
  7. Expandability.
  8. Generalized Activity.
  9. Expansion Relation
  10. Bounded Activity
  11. A Finite Acceptor for the Expansion Relation
  12. ω-Regular
  13. The Finiteness Problem for Bounded Automaton Semigroups

Key Result

Proposition 3.5

A word $x \in \Sigma^*$ expands a word $w \in \Sigma^*$ if and only if there are some $\bm{p}_1, \bm{p}_2 \in Q^+$ with $\bm{p}_1 \mathrel{\mathcal{E}_w} \bm{p}_2$ and $\bm{p}_1 \circ x \neq \bm{p}_2 \circ x$.

Theorems & Definitions (34)

  • Remark 2.1
  • Remark 2.2
  • Remark
  • Example 2.3
  • Remark
  • Remark
  • proof
  • proof
  • Example 2.7
  • Remark 2.8
  • ...and 24 more