A word ω is called synchronizing (recurrent, reset, directable) word of deterministic finite automaton (DFA) if ω brings all states of the automaton to an unique state. Černý 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 Schützenberger. 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. http://www.cs.biu.ac.il/~trakht/syn.html © Springer-Verlag Berlin Heidelberg 2007.
CITATION STYLE
Trahtman, A. N. (2007). Synchronization of some DFA. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 4484 LNCS, pp. 234–243). Springer Verlag. https://doi.org/10.1007/978-3-540-72504-6_21
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