The Synchronizing Probability Function of an Automaton

We study the synchronization phenomenon for deterministic finite state automata and the related longstanding Černý conjecture. We formulate this conjecture in the setting of a two-player probabilistic game. Our goal is twofold. On the one hand, the probabilistic interpretation is of interest in its own right and can be applied to real-world situations. On the other hand, our formulation makes use of standard convex optimization techniques, which appear powerful to shed light on Černý’s conjecture. We analyze the synchronization phenomenon through this particular point of view. Among other properties, we prove that the synchronization process cannot stagnate too long in a certain sense. We propose a new conjecture and demonstrate that its validity would imply Černý’s conjecture. We show numerical evidence for the pertinence of the approach.