Lower Bounds for Synchronizing Word Lengths in Partial Automata

TL;DR

This paper analyzes the minimal lengths of synchronizing words in partial automata (PFAs) with up to 7 states, extends known bounds, and constructs new PFAs with exponential synchronization lengths, surpassing classical bounds for DFAs.

Contribution

It provides a complete analysis of synchronizing word lengths for PFAs up to 7 states, introduces new constructions with exponential lengths, and compares these with DFA bounds.

Findings

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

More symbols in PFAs can increase the minimal synchronizing word length.

Constructed PFAs with exponential shortest synchronizing words, improving previous bounds.

Abstract

It was conjectured by \v{C}ern\'y 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 $n \leq 5$, and with bounds on the number of symbols for $n \leq 12$. Here we give the full analysis for $n \leq 7$, without bounds on the number of symbols. For PFAs (partial automata) on $\leq 7$ states we do a similar analysis as for DFAs and find the maximal shortest synchronizing word lengths, exceeding $(n-1)^2$ for $n \geq 4$. Where DFAs with long synchronization typically have very few symbols, for PFAs we observe that more symbols may increase the synchronizing word length. For PFAs on $\leq 10$ states and two symbols we investigate all occurring synchronizing word lengths. We give series of PFAs on two and three symbols, reaching the maximal possible length for some small values of $n$. For $n=6,7,8,9$, the construction on two symbols is the unique one reaching the maximal length. For both series the growth is faster than $(n-1)^2$, although still quadratic. Based on string rewriting, for arbitrary size we 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. Both PFAs are transitive. Finally, we show that exponential lengths are even possible with just one single undefined transition, again with transitive constructions.