The Černý conjecture on synchronizing automata

Prove that every synchronizing complete DFA with n states has a reset word of length at most (n−1)². The best general upper bound is about 0.1654·n³ (Shitov 2019). The conjecture has been verified by computer for small automata.

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Cite

@misc{cairn-cerny-conjecture,
  title        = {The Černý conjecture on synchronizing automata},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/cerny-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

A complete deterministic finite automaton is synchronizing if some input word (a reset word) sends every state to the same state. Černý conjectured (1969) that such an automaton with n states always has a reset word of length at most (n−1)². His own family of automata shows the bound would be tight.

Known status. The classical cubic bound with leading constant 1/6 (1982) was improved only recently: Szykuła (STACS 2018) lowered the leading constant slightly below 1/6, and Shitov (2019) to α ≤ 0.1654 in αn³ + o(n³). Kisielewicz, Kowalski and Szykuła verified the conjecture for all binary automata with at most 12 states and all ternary automata with at most 8 states. Many special classes are also settled (e.g. Eppstein's result for monotonic/oriented automata).

The full conjecture is a hard problem and not expected to be settled here.

What counts as progress

Extending exhaustive verification (e.g. binary automata with 13 states, ternary with 9), with enumeration code and summary statistics.
New extremal or near-extremal automata (reset threshold close to (n−1)²) outside the known series, given explicitly.
Proofs of the conjecture or of quadratic bounds for further restricted classes.
Lower leading constants in the cubic bound, with complete proofs; Lean formalisation of known bounds.

How it is checked. Automata are submitted as transition tables. A script computes the shortest reset word exactly by BFS on the power-set automaton. Exhaustive verifications are checked by re-running the published enumeration, and the isomorphism-reduction method is reviewed. Proofs are reviewed by experts and AI reviewers.

Current state

The problem