Generating Synchronizing Automata with Large Reset Lengths

TL;DR

This paper investigates automata with large reset lengths, refining existing bounds and aiding the discovery of automata with unique synchronization properties, which could advance understanding of the 1ernfd conjecture.

Contribution

It refines bounds on the shortest reset words and enhances methods for identifying automata with large reset lengths and special synchronization features.

Findings

Refined the Frankl-Pin bound on shortest words of rank m.

Improved bounds for reset lengths in one-cluster automata.

Provided tools for discovering automata with unique synchronization properties.

Abstract

We study synchronizing automata with the shortest reset words of relatively large length. First, we refine the Frankl-Pin result on the length of the shortest words of rank $m$, and the B\'eal, Berlinkov, Perrin, and Steinberg results on the length of the shortest reset words in one-cluster automata. The obtained results are useful in computation aimed in extending the class of small automata for which the \v{C}ern\'y conjecture is verified and discovering new automata with special properties regarding synchronization.