A characterization of those automata that structurally generate finite groups

Antonenko and Russyev independently have shown that any Mealy automaton with no cycles with exit–that is, where every cycle in the underlying directed graph is a sink component–generates a fi- nite (semi)group, regardless of the choice of the production functions. Antonenko has proved that this constitutes a characterization in the non-invertible case and asked for the invertible case, which is proved in this paper.

💡 Research Summary

**
The paper addresses the finiteness problem for groups generated by Mealy automata, focusing on the structural condition known as “no cycles with exit” – i.e., every directed cycle in the underlying graph is a sink component. Antonenko and Russyev independently proved that any Mealy automaton satisfying this condition generates a finite semigroup (and, in the invertible case, a finite group) regardless of the choice of production functions. Antonenko also showed that this condition is maximal in the non‑invertible setting: if an automaton contains a cycle with an exit, one can choose highly non‑invertible production functions so that the resulting semigroup is infinite. However, the invertible (i.e., reversible) case remained open.

The authors fill this gap by proving that the same maximality holds for invertible Mealy automata. Their main theorem states: Any automaton that contains a cycle with an exit can be enriched with invertible production functions to obtain an invertible Mealy automaton that generates an infinite group. Consequently, the property “all cycles are sink components” is not only sufficient but also necessary for finiteness of the generated group, even when the automaton is required to be invertible.

The proof strategy departs from the earlier non‑invertible arguments and relies heavily on the dual automaton (obtained by swapping states and alphabet) and a pumping lemma for two‑state reversible automata. The dual perspective allows the authors to translate structural properties of the original automaton into properties of a (often much smaller) reversible automaton, where classical group‑theoretic tools apply.

The paper proceeds as follows:

1. Preliminaries – Formal definitions of Mealy automata, reversibility, invertibility, bireversibility, and the associated (semi)groups. Several known facts (F1–F4) are recalled: pruning unreachable states does not affect finiteness, finiteness of a group equals finiteness of its semigroup for invertible automata, finiteness is preserved under dualisation, and any reversible but non‑bireversible automaton generates an infinite group.

2. Cycles with Exit – Precise definitions of external and internal exits for a directed cycle. An external exit is a transition from a cycle state to a vertex outside the cycle; an internal exit stays within the cycle but leads to a different successor than the one prescribed by the cycle’s label.

3. Pumping Lemma (Lemma 2) – For a reversible Mealy automaton with exactly two states, the generated semigroup is infinite iff for every integer N there exists a word u of length ≥ N such that the states u·x and u·y belong to the same connected component of the (|u|+1)-th power of the automaton. This lemma is the technical core that enables the authors to turn long paths in the dual automaton into cycles, thereby producing elements of infinite order.

4. Three Key Lemmas – Each handles a distinct configuration where a cycle with exit appears:

   • Lemma 3 (Binary alphabet, external exit): By assigning a transposition to the exit state and identity to all others, the dual automaton becomes a two‑state reversible automaton. Applying the pumping lemma yields an infinite group.
   • Lemma 4 (River‑of‑no‑return): When a cycle has an external exit to a state that cannot reach the cycle again, a construction mimicking the classical adding‑machine is used. A transposition on the exit’s label produces an element of infinite order.
   • Lemma 5 (Reversible automaton with exit): If the automaton is already reversible, one can find two distinct states that share a successor. By giving them distinct permutations while keeping all other states identity, the resulting automaton is invertible but not bireversible, and (F4) guarantees an infinite group.

5. Main Theorem (Theorem 6) – The authors combine the lemmas to cover all possibilities. After pruning unreachable states (F1), if the automaton is reversible they invoke Lemma 5. If it is not reversible, they locate either an external exit leading to a non‑reachable cycle (Lemma 4) or a situation where a path labelled by a single letter loops back to a state distinct from the start (producing a cycle with external exit). In the latter case Lemma 3 applies after suitable choice of permutations on the two involved letters. Thus any automaton with a cycle that has an exit can be enriched to generate an infinite group.

The paper’s contributions are twofold. First, it provides a complete structural characterisation of finiteness for groups generated by Mealy automata: the absence of cycles with exit is both necessary and sufficient, even under the stringent requirement of invertibility. Second, it introduces a methodological framework based on dualisation and a pumping lemma that may be useful for other decision problems in automaton groups, such as growth analysis or the order problem.

Overall, the work resolves an open question posed by Antonenko, closes the gap between the non‑invertible and invertible settings, and strengthens the understanding of how the underlying graph structure of a Mealy automaton dictates the algebraic nature of the group it defines.