The Finiteness Problem for Automaton Semigroups of Extended Bounded Activity

TL;DR

This paper extends the concept of activity in automaton semigroups to a broader setting, demonstrating that the finiteness problem for these structures with bounded activity is decidable using automata-theoretic methods.

Contribution

It introduces a generalized activity notion for automaton semigroups and proves the decidability of the finiteness problem for bounded activity cases, including subsemigroups.

Findings

Language of infinite orbits is a deterministic B"uchi language

Finiteness problem is decidable for bounded automaton semigroups

Decidability extends to sub-orbits under regular, suffix-closed languages

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 $\omega$-words with infinite orbits is effectively a deterministic B\"uchi 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 B\"uchi 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.