Modifying the upper bound on the length of minimal synchronizing word

Authors:
A. N. Trahtman
Affiliations:
Bar-Ilan University, Dep. of Math., Ramat Gan, Israel
Venue:
FCT'11 Proceedings of the 18th international conference on Fundamentals of computation theory
Year:
2011

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Abstract

A word w is called synchronizing (recurrent, reset, magic, directable) word of deterministic finite automaton (DFA) if w sends all states of the automaton to a unique state. In 1964 Jan Černy found a sequence of n-state complete DFA possessing a minimal synchronizing word of length (n - 1)2. He conjectured that it is an upper bound on the length of such words for complete DFA. Nevertheless, the best upper bound (n3 - n)/6 was found almost 30 years ago. We reduce the upper bound on the length of the minimal synchronizing word to n(7n2 +6n - 16)/48. An implemented algorithm for finding synchronizing word with restricted upper bound is described. The work presents the distribution of all synchronizing automata of small size according to the length of an almost minimal synchronizing word.