DFAs and PFAs with Long Shortest Synchronizing Word Length

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), and he gave a sequence of DFAs for which this bound is reached. Until now a full analysis of all DFAs reaching this bound was only given for n ≤ 4, and with bounds on the number of symbols for n ≤ 10. Here we give the full analysis for n ≤ 6, without bounds on the number of symbols. For PFAs the bound is much higher. For n ≤ 6 we do a similar analysis as for DFAs and find the maximal shortest synchronizing word lengths, exceeding (n − 1) for n = 4, 5, 6. For arbitrary n we give a construction of a PFA on three symbols with exponential shortest synchronizing word length, giving significantly better bounds than earlier exponential constructions. We give a transformation of this PFA to a PFA on two symbols keeping exponential shortest synchronizing word length, yielding a better bound than applying a similar known transformation.