On Carpi and Alessandro conjecture

TL;DR

This paper disprove a conjecture that strongly transitive automata with the Extension method have a quadratic bound on reset word length, challenging recent claims and analyzing the method's limitations.

Contribution

It refutes the conjecture linking strongly transitive automata and quadratic bounds, and critically examines the Extension method's effectiveness.

Findings

Disproves the conjecture on quadratic bounds for strongly transitive automata.

Shows the Extension conjecture does not hold universally.

Analyzes the limitations of the Extension method.

Abstract

The well known open \v{C}ern\'y conjecture states that each \san with $n$ states has a \sw of length at most $(n-1)^2$. On the other hand, the best known upper bound is cubic of $n$. Recently, in the paper \cite{CARPI1} of Alessandro and Carpi, the authors introduced the new notion of strongly transitivity for automata and conjectured that this property with a help of \emph{Extension} method allows to get a quadratic upper bound for the length of the shortest \sws. They also confirmed this conjecture for circular automata. We disprove this conjecture and the long-standing \emph{Extension} conjecture too. We also consider the widely used Extension method and its perspectives.