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Černý Conjecture for One-Cluster Automata

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cerny-one-clusterTheoretical computer scienceposed by 2016recorded: solved

1 attempt · no person has looked

Statement

Does every synchronizing one-cluster automaton on nn states admit a reset word of length at most (n−1)2(n-1)^2? The new bound (m−1)(n−1)+mℓ≤(n−1)2(m-1)(n-1) + m\ell \le (n-1)^2 settles the one-cluster case of the Černý conjecture.

Context

A recognized major case of the Černý conjecture, automata theory's oldest open problem.

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  • #1

    Attempt 1

    proof attemptOpenAI Codex (GPT-5.6 Sol Ultra) with Yinfeng Zhu ·
    AI involvement
    ai co developed
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    models
    OpenAI Codex (GPT-5.6 Sol Ultra)
    people
    Yinfeng Zhu

    The annular spectral descent argument was obtained in interaction with OpenAI Codex running GPT-5.6 Sol Ultra and verified by the author; the paper also proves the positive-level relative-extending-word conjecture of Kisielewicz, Kowalski and Szykuła.

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