Matrix approach to synchronizing automata

TL;DR

This paper explores a matrix-based approach to studying synchronizing automata, focusing on the Cerny conjecture, which posits a quadratic upper bound on the length of minimal synchronizing words for finite automata.

Contribution

It introduces a novel matrix framework for analyzing synchronizing automata, providing new insights into the Cerny conjecture and automata synchronization.

Findings

Matrix approach offers new tools for automata analysis

Supports the quadratic bound hypothesis for minimal synchronizing words

Provides a basis for further research into automata synchronization

Abstract

A word $w$ of letters on edges of underlying graph $\Gamma$ of deterministic finite automaton (DFA) is called synchronizing if $w$ sends all states of the automaton to a unique state. J. \v{C}erny discovered in 1964 a sequence of $n$-state complete DFA possessing a minimal synchronizing word of length $(n-1)^2$. The hypothesis, well known today as \v{C}erny conjecture, claims that $(n-1)^2$ is a precise upper bound on the length of such a word over alphabet $\Sigma$ of letters on edges of $\Gamma$ for every complete $n$-state DFA. The hypothesis was formulated distinctly in 1966 by Starke. A special classes of matrices induced by words in the alphabet of labels on edges of the underlying graph of DFA are used for the study of synchronizing automata.