On the Synchronizing Probability Function and the Triple Rendezvous Time for Synchronizing Automata

We push further a recently proposed approach for studying synchronizing automata and Cerny's conjecture, namely, the synchronizing probability function. In this approach, the synchronizing phenomenon is reinterpreted as a Two-Player game, in which the optimal strategies of the players can be obtained through a Linear Program. Our analysis mainly focuses on the concept of triple rendezvous time, the length of the shortest word mapping three states onto a single one. It represents an intermediate step in the synchronizing process, and is a good proxy of its overall length. Our contribution is twofold. First, using the synchronizing probability function and properties of linear programming, we provide a new upper bound on the triple rendezvous time. Second, we disprove a conjecture on the synchronizing probability function by exhibiting a family of counterexamples. We discuss the game theoretic approach and possible further work in the light of our results.