List of Results on the \v{C}ern\'y Conjecture and Reset Thresholds for Synchronizing Automata

TL;DR

This survey compiles known results and open problems related to the cernfd conjecture and reset thresholds in finite automata, highlighting classes where the conjecture is proven or remains unresolved.

Contribution

It provides a comprehensive overview of the current state of research on the cernfd conjecture and reset thresholds, including recent advances and open questions.

Findings

The cernfd conjecture is proven for several classes of automata.

Quadratic upper bounds on reset words are known for some classes where the conjecture is open.

The survey reflects the state of the art as of January 13, 2026.

Abstract

We survey results in the literature that establish the \v{C}ern\'y conjecture for various classes of finite automata. We also list classes for which the conjecture remains open, but a quadratic (in the number of states) upper bound on the minimum length of reset words is known. The results presented reflect the state of the art as of January 13, 2026.