On Carpi and Alessandro conjecture

Abstract

The well known open Černý 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.

arXiv comment11 pages, 1 figure