Slowly synchronizing automata with fixed alphabet size

Abstract

It was conjectured by Černý in 1964 that a synchronizing DFA on n states always has a shortest synchronizing word of length at most (n−1)2, and he gave a sequence of DFAs reaching this bound.In this paper, we investigate the role of the alphabet size. For each possible alphabet size, we count DFAs on n≤6 states which synchronize in (n−1)2−e steps, for all e<2⌈n/2⌉. Furthermore, we give constructions of automata with any number of states, and 3, 4, or 5 symbols, which synchronize slowly, namely in n2−3n+O(1) steps.In addition, our results prove Černý's conjecture for n≤6. Our computations lead to 31 DFAs on 3, 4, 5 or 6 states, which synchronize in (n−1)2 steps. Of these DFA's, 19 are new. The remaining 12, which were already known, are exactly the minimal ones.The 19 new DFAs are extensions of automata which were already known. But for n>3, we prove that the Černý automaton on n states does not admit non-trivial extensions with the same smallest synchronizing word length (n−1)2.