The Černy Conjecture for Aperiodic Automata

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. Černý conjectured in 1964 that every n-state synchronizable DFA possesses a synchronizing word of length at most (n−1). 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 Černý conjecture holds true.