DFAs and PFAs with Long Shortest Synchronizing Word Length

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 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 the bound is much higher. For $n\le 6$ 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 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.

16 pages, 2 figures source code added

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