Synchronizing non-deterministic finite automata

TL;DR

This paper demonstrates that non-deterministic finite automata can be synchronized similarly to deterministic automata, extending Cerny's conjecture and establishing bounds for various classes, with implications for identifying critical automata.

Contribution

It shows a unique mapping between CNFAs and DFAs preserving synchronizing properties, extending Cerny's conjecture and bounds to non-deterministic automata.

Findings

Cerny's conjecture generalizes to CNFAs

Established sharper bounds for certain CNFA classes

Identified all critical CNFAs with up to 6 states

Abstract

In this paper, we show that every D3-directing CNFA can be mapped uniquely to a DFA with the same synchronizing word length. This implies that \v{C}ern\'y's conjecture generalizes to CNFAs and that the general upper bound for the length of a shortest D3-directing word is equal to the Pin-Frankl bound for DFAs. As a second consequence, for several classes of CNFAs sharper bounds are established. Finally, our results allow us to detect all critical CNFAs on at most 6 states. It turns out that only very few critical CNFAs exist.