Trahtman, Avraham N. The Černý conjecture for aperiodic automata. (English) Zbl 1152.68461 Discrete Math. Theor. Comput. Sci. 9, No. 2, 3-10 (2007). 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. Cited in 40 Documents MSC: 68Q45 Formal languages and automata PDFBibTeX XMLCite \textit{A. N. Trahtman}, Discrete Math. Theor. Comput. Sci. 9, No. 2, 3--10 (2007; Zbl 1152.68461) Full Text: arXiv Link