Synchronization of Some DFA

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A word w is called synchronizing (recurrent, reset, directable) word of deterministic ﬁnite automaton (DFA) if w brings all states of the automaton to an unique state. ˇCerny conjectured in 1964 that every nstate synchronizable automaton possesses a synchronizing word of length at most (n − 1)2 . The problem is still open. It will be proved that the minimal length of synchronizing word is not greater than (n − 1)2 /2 for every n-state (n > 2) synchronizable DFA with transition monoid having only trivial subgroups (such automata are called aperiodic). This important class of DFA accepting precisely star-free languages was involved and studied by Sch˘utzenberger. So for aperiodic automata as well as for automata accepting only star-free languages, the ˇCern´y conjecture holds true. Some properties of an arbitrary synchronizable DFA and its transition semigroup were established. http://www.cs.biu.ac.il/∼trakht/syn.html

A. N. Trahtman

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