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.
Highlights
The problem of synchronization of deterministic finite automaton (DFA) is natural and various aspects of this problem were touched upon the literature
Cernyfound in 1964 [2] an n-state DFA whose shortest synchronizing word was of length (n− 1)2
He conjectured that this is the maximum length of the shortest synchronizing word for any DFA with n states
Summary
The problem of synchronization of DFA is natural and various aspects of this problem were touched upon the literature. Cernyfound in 1964 [2] an n-state DFA whose shortest synchronizing word was of length (n− 1). The best upper bound for the length of the shortest synchronizing word for DFA with n states known so far is equal to (n3 − n)/6 [5, 7, 9]. The existence of some non-trivial subgroup in the transition semigroup of the automaton is essential in many investigations of the Cernyconjecture, see, e.g., [3, 8]. We prove that every n-state aperiodic DFA with a state that is accessible from every state of the automaton has a synchronizing word of length not greater than n(n − 1)/2, and for aperiodic automata as well as for automata accepting only star-free languages, the Cernyconjecture holds true. In the case when the underlying graph of the aperiodic DFA is strongly connected, this upper bound has been recently improved by Volkov who has reduced the estimation to n(n + 1)/6
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