Synchronization of Some DFA
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. Černy conjectured in 1964 that every n-state 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 Schŭtzenberger. So for aperiodic automata as well as for automata accepting only star-free languages, the Černý conjecture holds true.
Some properties of an arbitrary synchronizable DFA and its transition semigroup were established.
KeywordsDeterministic finite automata synchronization aperiodic semigroup Černy conjecture
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