Notable trends concerning the synchronization of graphs and automata

Abstract

A word w is called synchronizing (recurrent, reset, directed) word of a deterministic finite automaton (DFA) if w sends all states of the automaton on a unique state. Jan Cerny had found in 1964 a sequence of n-state complete DFA with shortest synchronizing word of length (n− 1). He had conjectured that it is an upper bound for the length of the shortest synchronizing word for any n-state complete DFA. This simply looking conjecture is now one of the most longstanding open problems in the theory of finite automata. Moreover, the examples of DFA with shortest synchronizing word of length (n − 1) are relatively rare. To the Cerny sequence were added in all examples of Cerny, Piricka and Rosenauerova (1971), of Kari (2001) and of Roman (2004). An effective program for search of automata with minimal reset word of relatively great length has studied all automata of size n ≤ 10 for q = 2 (q size of alphabet) and of size n ≤ 7 for q ≤ 4. There are no contradictory examples for the Cerny conjecture in this class of automata. Moreover, the program does not find new examples of DFA with reset word of length (n−1)2 for automata with n > 4 as well as for q > 3. New examples of DFA with shortest synchronizing word of length (n − 1) were found only for n = 3, 4 and for q = 2, 3. And what is more, the examples with minimal length of reset word disappear even for values near the Cerny bound (n− 1) with growth of the size of the automaton as well as of the size of the alphabet. The gap between (n − 1) and the nearest of the minimal lengths of reset word appears for n = 6. The following table displays this interesting trend for the length of minimal reset words less than (n− 1).