On Completely Reachable Automata and Subset Reachability

TL;DR

This paper investigates subset reachability in synchronizing automata, presenting families with unreachable subsets and exponential reachability lengths, and analyzes completely reachable automata with counterexamples and a revised conjecture.

Contribution

It introduces new families of automata challenging existing conjectures and provides a counterexample and an alternative proof regarding completely reachable automata.

Findings

Some subsets require exponentially long words to reach.

Families of automata do not satisfy Don's Conjecture.

Counterexample to the $ ext{Gamma}_1$-graph conjecture and an alternative proof.

Abstract

This article focuses on subset reachability in synchronizing automata. First, we provide families of synchronizing automata with subsets which cannot be reached with short words. These families do not fulfil Don's Conjecture about subset reachability. Moreover, they show that some subsets need exponentially long words to be reached, and that the restriction of the conjecture to included subsets also does not hold. Second, we analyze completely reachable automata and provide a counterexample to the conjecture of Bondar and Volkov about the so-called $\Gamma_1$-graph. We finally prove an alternative version of this conjecture.