The Černý conjecture on synchronizing automata

Show that every synchronizing automaton with n states has a reset word of length at most (n − 1)².

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State the problem, from Formal Conjectures

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	1	+# The Černý conjecture on synchronizing automata
	2	+
	3	+A finite automaton is synchronizing if some sequence of inputs drives it into one and the same state wherever it starts, like a reset button for a machine whose state is unknown. Černý conjectured that for an automaton with $n$ states, a reset sequence of length $(n-1)^2$ always suffices, which would be exactly tight. More than sixty years later the best general bound is still cubic in $n$.
	4	+
	5	+## Problem
	6	+
	7	+A complete deterministic finite automaton consists of a finite set of states $Q$ with $\|Q\| = n$, a finite input alphabet $\Sigma$, and a transition function $\delta : Q \times \Sigma \to Q$, extended to words letter by letter. A word $w$ is synchronizing (or a reset word) if $\delta(q, w)$ is the same state for every $q \in Q$, and the automaton is synchronizing if it has such a word.
	8	+
	9	+Conjecture (Černý). Every synchronizing automaton with $n$ states has a synchronizing word of length at most $(n - 1)^2$.
	10	+
	11	+## Background
	12	+
	13	+Ján Černý (1964) described, for each $n$, an automaton whose shortest synchronizing word has length exactly $(n-1)^2$, and the conjecture that no automaton is worse is attributed to him in the 1960s. It is one of the best-known open problems in automata theory: easy to state, testable by computer for small $n$, and resistant to general methods. Volkov's survey describes its history and the special classes of automata for which it is known.
	14	+
	15	+## What is known
	16	+
	17	+- Every synchronizing automaton with $n$ states has a reset word of length at most $\alpha n^3 + o(n^3)$ for an absolute constant $\alpha$. Pin and Frankl (1982) gave $\alpha = 1/6$; Szykuła (2017) lowered it to about $0.1664$, and Shitov (2019) to $\alpha \le 0.1654$, the best bound known.
	18	+- The Černý automata show that $(n-1)^2$ cannot be lowered.
	19	+- The bound $(n-1)^2$ has been verified for small $n$ and proved for special classes, for example Eulerian, aperiodic and cyclic automata, as recorded in the formal-conjectures file. A July 2026 preprint (Zhu, unrefereed, obtained with AI assistance) claims it for one-cluster automata.
	20	+
	21	+## What would count as progress
	22	+
	23	+- A counterexample: an automaton with $n$ states whose shortest reset word is longer than $(n-1)^2$. Its reset threshold can be computed exactly by breadth-first search over subsets of states, so any candidate is checkable.
	24	+- A better constant in the cubic bound, or a bound of order $n^{3 - \varepsilon}$.
	25	+- Proofs of the quadratic bound for further natural classes of automata, or extended exhaustive verification for small $n$ and alphabet sizes, with published code.
	26	+- A formal proof of the Lean statement `cerny_conjecture`, or of Shitov's bound stated in the same file, [Wikipedia/CernyConjecture.lean](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/CernyConjecture.lean).
	27	+
	28	+## References
	29	+
	30	+1. J. Černý, [Poznámka k homogénnym experimentom s konečnými automatmi](https://dml.cz/bitstream/handle/10338.dmlcz/126647/MathSlov_14-1964-3_2.pdf), Mat.-fyz. časopis 14 (1964), 208–216.
	31	+2. M. V. Volkov, [Synchronizing automata and the Černý conjecture](https://doi.org/10.1007/978-3-540-88282-4_4), in Language and Automata Theory and Applications (LATA 2008), LNCS 5196, 11–27.
	32	+3. Y. Shitov, [An improvement to a recent upper bound for synchronizing words of finite automata](https://arxiv.org/abs/1901.06542), J. Autom. Lang. Comb. 24 (2019), 367–373.
	33	+4. Y. Zhu, [The Černý conjecture for one-cluster automata via annular spectral descent](https://arxiv.org/abs/2607.19675), arXiv:2607.19675 (2026).
	34	+
	35	+## Source
	36	+
	37	+Imported from [Formal Conjectures Wikipedia/CernyConjecture](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/CernyConjecture.lean), curated by Google DeepMind and contributors. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.