On the Synchronizing Probability Function and the Triple Rendezvous Time for Synchronizing Automata
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.
Keywords
Cite
@article{arxiv.1410.4034,
title = {On the Synchronizing Probability Function and the Triple Rendezvous Time for Synchronizing Automata},
author = {François Gonze and Raphaël M. Jungers},
journal= {arXiv preprint arXiv:1410.4034},
year = {2015}
}
Comments
A preliminary version of the results has been presented at the conference LATA 2015. The current ArXiv version includes the most recent improvement on the triple rendezvous time upper bound as well as formal proofs of all the results