Synchronizing Automata with Extremal Properties

TL;DR

This paper introduces new classes of synchronizing automata with extremal properties, disproves a conjecture on subset extendability, and presents automata with quadratic reset word lengths over larger alphabets.

Contribution

It provides counterexamples to existing conjectures, introduces the image-extension conjecture, and constructs new automata series with quadratic reset lengths over ternary alphabets.

Findings

Disproves the conjecture that all subsets are $cn$-extendable in strongly connected automata.

Shows automata with subsets requiring words of length $n^2/4+O(n)$ to extend.

Provides the first examples of slowly irreducibly synchronizing automata over a ternary alphabet.

Abstract

We present a few classes of synchronizing automata exhibiting certain extremal properties with regard to synchronization. The first is a series of automata with subsets whose shortest extending words are of length $\varTheta(n^2)$, where $n$ is the number of states of the automaton. This disproves a conjecture that every subset in a strongly connected synchronizing automaton is $cn$-extendable, for some constant $c$, and in particular, shows that the cubic upper bound on the length of the shortest reset words cannot be improved generally by means of the extension method. A detailed analysis shows that the automata in the series have subsets that require words as long as $n^2/4+O(n)$ in order to be extended by at least one element. We also discuss possible relaxations of the conjecture, and propose the image-extension conjecture, which would lead to a quadratic upper bound on the length of the shortest reset words. In this regard we present another class of automata, which turn out to be counterexamples to a key claim in a recent attempt to improve the Pin-Frankl bound for reset words. Finally, we present two new series of slowly irreducibly synchronizing automata over a ternary alphabet, whose lengths of the shortest reset words are $n^2-3n+3$ and $n^2-3n+2$, respectively. These are the first examples of such series of automata for alphabets of size larger than two.