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arXiv · 1703.07618

DFAs and PFAs with Long Shortest Synchronizing Word Length

Michiel de Bondt·Henk Don·Hans Zantema
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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 $n \leq 4$, and with bounds on the number of symbols for $n \leq 10$. Here we give the full analysis for $n \leq 6$, without bounds on the number of symbols. For PFAs the bound is much higher. For $n \leq 6$ 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.

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Michiel de Bondt, Henk Don, Hans Zantema. 2017-06-16. DFAs and PFAs with Long Shortest Synchronizing Word Length. https://arxiv.org/abs/1703.07618

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