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.
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 2026-11-12}
} Also: CITATION.cff · Atom feed of results
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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.
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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.
History of the bounds
| Year | Result | By |
|---|---|---|
| 1983 | ≤ (n³−n)/6 | Pin; Frankl |
| 2018 | ≈ 0.1664·n³ | Szykuła |
| 2019 | ≈ 0.1654·n³ | Shitov |