Experiments with Synchronizing Automata

TL;DR

This paper enhances algorithms for generating synchronizing automata, enabling verification of the cernfd conjecture for small automata, and discusses implications of these findings.

Contribution

It introduces improved algorithms and bounds for synchronizing automata, facilitating the verification of the cernfd conjecture for small state automata.

Findings

Verified the cernfd conjecture for all binary automata with up to 12 states.

Verified the cernfd conjecture for all ternary automata with up to 8 states.

Improved bounds allow analysis of automata with larger state spaces.

Abstract

We have improved an algorithm generating synchronizing automata with a large length of the shortest reset words. This has been done by refining some known results concerning bounds on the reset length. Our improvements make possible to consider a number of conjectures and open questions concerning synchronizing automata, checking them for automata with a small number of states and discussing the results. In particular, we have verified the \v{C}ern\'y conjecture for all binary automata with at most 12 states, and all ternary automata with at most 8 states.