Open problemChecked July 24, 2026

Černý Conjecture

The main length bound conjecture for synchronizing finite automata.

Computer science Automata theory Combinatorics

Mathematical statement

Every synchronizing deterministic finite automaton with $n$ states has a synchronizing word of length at most $(n-1)^2$ .

Statement source: Therefore editorial statement

Statement terms: Source-specific

Editorial wording by Therefore. No statement reuse license is granted.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

cerny_conjecture

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem cerny_conjecture :    answer(sorry) ↔      ∀ {α σ : Type*} [Fintype σ]        (M : DFA α σ) (hM : M.IsSynchronizing),        ∃ w : List α,          M.IsSynchronizingWord w ∧          w.length ≤ (Fintype.card σ - 1)^2 := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Černý conjecture overviewbackground Černý's original 1964 paperoriginal