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The biased circular automata which verify Černý’s conjecture. (Les automates circulaires biaisés vérifient la conjecture de Černý.) (French) Zbl 0877.68083

Summary: An action \(X\times A^+\to X: (x,w)\mapsto x*w\) of the free semigroup \(A^+\) over the alphabet \(A\) is synchronized by a word \(w\in A^+\) if \(x*w= y*w\) for all \(x,y\in X\). Černý conjectured that if there is such a \(w\) and \(|X|=n\), then there is one of length \(\leq(n-1)^2\). The conjecture remains open, except for particular classes of actions. An action is circular if there is \(c\in A\) acting as a circular permutation, and biased if there is \(b\in A\) with only one \(x\in X\) such that \(|x*b^{-1}|\geq 2\). We confirm the Černý conjecture for biased circular actions.

MSC:

68Q45 Formal languages and automata
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References:

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