A quadratic upper bound on the reset thresholds of synchronizing automata containing a transitive permutation group

TL;DR

This paper establishes a quadratic upper bound on the reset threshold for certain synchronizing automata with a transitive permutation group, improving understanding of their synchronization properties.

Contribution

It proves a quadratic upper bound on reset thresholds for automata with transitive permutation groups and large image actions, extending previous bounds for specific automata classes.

Findings

Existence of a synchronizing word of length at most 2n^2 - 7n + 7

Applicable to automata with transitive permutation groups and large image actions

Advances bounds on the Cerný conjecture for specific automata classes

Abstract

For any synchronizing $n$-state deterministic automaton, \v{C}ern\'{y} conjectures the existence of a synchronizing word of length at most $(n-1)^2$. We prove that there exists a synchronizing word of length at most $2n^2 - 7n + 7$ for every synchronizing $n$-state deterministic automaton that satisfies the following two properties: 1. The image of the action of each letter contains at least $n-1$ states; 2. The actions of bijective letters generate a transitive permutation group on the state set.