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Reset threshold bound for strongly connected aperiodic automata

Prove that the reset threshold of every strongly connected aperiodic deterministic finite automaton with n states is at most n−1.

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Background

Aperiodic automata are those whose transition monoids have no nontrivial subgroups. Trahtman proved general quadratic bounds for aperiodic automata, and Volkov improved bounds in certain strongly connected cases.

The authors report a conjecture that for strongly connected aperiodic automata the reset threshold is bounded linearly by n−1, supported by exhaustive searches for small sizes, but it remains unproved in full generality.

References

No examples of strongly connected aperiodic n-state automata are available with reset threshold higher than n-1, and it is conjectured that the reset threshold for such automata does not exceed n-1.

— List of Results on the Černý Conjecture and Reset Thresholds for Synchronizing Automata (2508.15655 - Volkov, 21 Aug 2025) in Section B2, “Aperiodic automata,” Lower bound