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The Černý conjecture for aperiodic automata. (English) Zbl 1152.68461
Summary: 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.
MSC:
68Q45Formal languages and automata