### The Černý Conjecture for Aperiodic Automata

*Avraham N. Trahtman*

#### Abstract

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.Full Text: PDF PostScript