The Černý conjecture for aperiodic automata

Avraham N. Trahtman (1)

1 Department of Mathematics [Israël]

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.

Domaines: Mathématique discrète [cs.DM]

Avraham N. Trahtman. The Černý conjecture for aperiodic automata. Discrete Mathematics and Theoretical Computer Science, 2007, Vol. 9 no. 2 (2), pp.3--10. ⟨10.46298/dmtcs.395⟩. ⟨hal-00966534⟩