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 synchronizing word of length at most \((n-1)^2\), 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 \(n \le 4\), and with bounds on the number of symbols for \(n \le 10\). Here we give the full analysis for \(n \le 6\), without bounds on the number of symbols.
For PFAs on \(n\le 6\) states we do a similar analysis as for DFAs and find the maximal shortest synchronizing word lengths, exceeding \((n-1)^2\) for \(n =4,5,6\). For arbitrary n we use rewrite systems to construct 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.
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