Finding DFAs with Maximal Shortest Synchronizing Word Length

Abstract

It was conjectured by Cerný 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 for which this bound is reached. In 2006 Trahtman conjectured that apart from Cerný’s sequence only 8 DFAs exist attaining the bound. He gave an investigation of all DFAs up to certain size for which the bound is reached, and which do not contain other synchronizing DFAs. Here we extend this analysis in two ways: we drop this latter condition, and we drop limits on alphabet size. For \(n \le 4\) we do the full analysis yielding 19 new DFAs with smallest synchronizing word length \((n-1)^2\), refuting Trahtman’s conjecture. All these new DFAs are extensions of DFAs that were known before. For \(n \ge 5\) we prove that none of the DFAs in Trahtman’s analysis can be extended similarly. In particular, as a main result we prove that the Cerný examples \(C_n\) do not admit non-trivial extensions keeping the same smallest synchronizing word length \((n-1)^2\).