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

TL;DR

This paper introduces new bounds on the triple rendezvous time in automata and challenges existing beliefs about the synchronizing probability function, providing fresh insights into Cerny's conjecture.

Contribution

It presents a novel upper bound on triple rendezvous time and disproves a conjecture regarding the growth of the synchronizing probability function.

Findings

New upper bound on triple rendezvous time

Counterexamples disprove a conjecture on synchronizing probability function growth

Provides directions for future research on Cerny's conjecture

Abstract

Cerny's conjecture is a longstanding open problem in automata theory. We study two different concepts, which allow to approach it from a new angle. The first one is the triple rendezvous time, i.e., the length of the shortest word mapping three states onto a single one. The second one is the synchronizing probability function of an automaton, a recently introduced tool which reinterprets the synchronizing phenomenon as a two-player game, and allows to obtain optimal strategies through a Linear Program. Our contribution is twofold. First, by coupling two different novel approaches based on the synchronizing probability function and properties of linear programming, we obtain a new upper bound on the triple rendezvous time. Second, by exhibiting a family of counterexamples, we disprove a conjecture on the growth of the synchronizing probability function. We then suggest natural follow-ups towards Cernys conjecture.