×

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:

68Q45 Formal languages and automata
PDFBibTeX XMLCite
Full Text: arXiv Link