DFAs and PFAs with Long Shortest Synchronizing Word Length

TL;DR

This paper analyzes the maximum shortest synchronizing word lengths for DFAs and PFAs, providing full classifications for small cases and new constructions showing exponential lengths for PFAs, advancing understanding of synchronization bounds.

Contribution

It offers a complete analysis of DFAs with up to 6 states regarding their shortest synchronizing words and introduces new PFA constructions with exponential bounds.

Findings

DFAs with up to 6 states fully classified for maximum shortest synchronizing words.

PFAs can have significantly longer shortest synchronizing words than DFAs, exceeding quadratic bounds.

New PFA constructions achieve exponential shortest synchronizing word lengths, surpassing previous bounds.

Abstract

It was conjectured by \v{C}ern\'y 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.