Synchronizing Automata on Quasi Eulerian Digraph

TL;DR

This paper introduces a novel approach using Markov chains and Perron-Frobenius theory to establish quadratic upper bounds on reset lengths for a generalized class of automata, advancing the understanding of the Černý conjecture.

Contribution

It presents a new method for proving quadratic bounds on reset lengths, extending results to a broader class of automata beyond Eulerian digraphs.

Findings

Quadratic upper bound established for the generalized class of automata.

New approach connects automata theory with Markov chains and Perron-Frobenius theory.

Advances towards the Černý conjecture by broadening classes with known bounds.

Abstract

In 1964 \v{C}ern\'{y} conjectured that each $n$-state synchronizing automaton posesses a reset word of length at most $(n-1)^2$. From the other side the best known upper bound on the reset length (minimum length of reset words) is cubic in $n$. Thus the main problem here is to prove quadratic (in $n$) upper bounds. Since 1964, this problem has been solved for few special classes of \sa. One of this result is due to Kari \cite{Ka03} for automata with Eulerian digraphs. In this paper we introduce a new approach to prove quadratic upper bounds and explain it in terms of Markov chains and Perron-Frobenius theories. Using this approach we obtain a quadratic upper bound for a generalization of Eulerian automata.