A bound for the shortest reset words for semisimple synchronizing automata via the packing number

TL;DR

This paper establishes an upper bound on the length of the shortest reset words for semisimple synchronizing automata, linking automaton properties with combinatorial packing numbers to improve understanding of automaton synchronization.

Contribution

It introduces a novel bound for reset words based on the packing number, connecting automaton structure with combinatorial set theory.

Findings

Bound improves previous estimates for reset word lengths.

Uses combinatorial packing number to relate automaton properties.

Provides a new approach to analyze automaton synchronization complexity.

Abstract

We show that if a semisimple synchronizing automaton with $n$ states has a minimal reachable non-unary subset of cardinality $r\ge 2$, then there is a reset word of length at most $(n-1)D(2,r,n)$, where $D(2,r,n)$ is the $2$-packing number for families of $r$-subsets of $[1,n]$.