The Cerny conjecture for automata respecting intervals of a directed graph

TL;DR

This paper proves the Cerny conjecture for a class of automata that preserve interval properties of directed graphs, extending previous results and providing a broader understanding of synchronizing automata.

Contribution

It introduces a proof of the Cerny conjecture for automata respecting directed graph intervals, generalizing earlier specific cases.

Findings

Proves the Cerny conjecture for a new class of automata

Unifies previous partial results under a broader framework

Enhances understanding of automata synchronization properties

Abstract

The \v{C}ern\'y's conjecture states that for every synchronizing automaton with n states there exists a reset word of length not exceeding (n-11)^2. We prove this conjecture for a class of automata preserving certain properties of intervals of a directed graph. Our result unifies and generalizes some earlier results obtained by other authors.