On randomized generation of slowly synchronizing automata

TL;DR

This paper introduces a randomized method for generating automata with large reset thresholds, leveraging primitive matrix sets, and provides both empirical and theoretical insights into their properties.

Contribution

It presents a novel randomized procedure for creating automata with high reset thresholds and analyzes the properties of primitive matrix sets in this context.

Findings

Algorithm finds automata with larger reset thresholds than uniform random generation.

New automata families with reset threshold Ω(n^2/4) are identified.

Theoretical results show primitive sets of matrices have high probability of certain properties.

Abstract

Motivated by the randomized generation of slowly synchronizing automata, we study automata made of permutation letters and a merging letter of rank $ n\!-\!1 $. We present a constructive randomized procedure to generate synchronizing automata of that kind with (potentially) large alphabet size based on recent results on \textit{primitive} sets of matrices. We report numerical results showing that our algorithm finds automata with much larger reset threshold than a mere uniform random generation and we present new families of automata with reset threshold of $ \Omega(n^2/4) $. We finally report theoretical results on randomized generation of primitive sets of matrices: a set of permutation matrices with a $ 0 $ entry changed into a $ 1 $ is primitive and has exponent of $ O(n\log n) $ with high probability in case of uniform random distribution and the same holds for a random set of binary matrices where each entry is set, independently, equal to $ 1 $ with probability $ p $ and equal to $ 0 $ with probability $ 1-p $, when $ np-\log n\rightarrow\infty $ as $ n\rightarrow\infty $.