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The Černý conjecture for aperiodic automata

Abstract : A word w is called a synchronizing (recurrent, reset, directable) word of a deterministic finite automaton (DFA) if w brings all states of the automaton to some specific state; a DFA that has a synchronizing word is said to be synchronizable. Cerny conjectured in 1964 that every n-state synchronizable DFA possesses a synchronizing word of length at most (n-1)2. We consider automata with aperiodic transition monoid (such automata are called aperiodic). We show that every synchronizable n-state aperiodic DFA has a synchronizing word of length at most n(n-1)/2. Thus, for aperiodic automata as well as for automata accepting only star-free languages, the Cerny conjecture holds true.
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Avraham N. Trahtman. The Černý conjecture for aperiodic automata. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2007, 9 (2), pp.3--10. ⟨hal-00966534⟩

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