The Synchronizing Probability Function for Primitive Sets of Matrices

TL;DR

This paper introduces a probabilistic game framework and convex optimization methods to analyze primitive sets of matrices, aiming to address the cernfd conjecture and automata synchronization bounds.

Contribution

It formulates the primitivity process as a two-player game and develops tools for approximating the exponent, proposing a conjecture linking primitive sets to automata reset thresholds.

Findings

Developed a convex optimization-based approach to analyze primitivity.

Numerical results support a conjecture for quadratic bounds on automata reset thresholds.

Proposed a new perspective connecting matrix primitivity with automata synchronization.

Abstract

Motivated by recent results relating synchronizing DFAs and primitive sets, we tackle the synchronization process and the related longstanding \v{C}ern\'{y} 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 develop a tool for approximating and upper bounding the exponent of any primitive set and supported by numerical results we state a conjecture that, if true, would imply a quadratic upper bound on the reset threshold of a new class of automata.