Randomized trial demonstrates decidability of the finiteness problem in automaton semigroups, implying further implications for related mathematical constructs.
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. We show that the language of 𝜔-words with infinite orbits is effectively a deterministic Büchi language for automata with bounded extended activity, which yields decidability of the finiteness problem for complete automaton semigroups and monoids of bounded activity (solving an open problem by Bartholdi, Godin, Klimann and Picantin). In fact, we obtain a stronger result also covering finitely generated subsemigroups.
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D’Angeli et al. (2026) studied this question.
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