The Synchronizing Probability Function for Primitive Sets of Matrices

Abstract

Motivated by recent results relating synchronizing automata and primitive sets, we tackle the synchronization process and the related longstanding Cerný conjecture by studying the primitivity phenomenon for sets of nonnegative matrices having neither zero-rows nor zero-columns. We formulate the primitivity process in the setting of a two-player probabilistic game and we make use of convex optimization techniques to describe its behavior. We report numerical results and supported by them we state a conjecture that, if true, would imply an upper bound of \( n(n-1) \) on the reset threshold of a certain class of automata.