The Cerny Conjecture

TL;DR

This paper simplifies the Cerny conjecture, proves it for specific automata types, and establishes quadratic bounds using Markov chain theory and linear programming.

Contribution

It reduces the Cerny conjecture to a simpler form and proves it for one-cluster automata, providing new bounds and methods.

Findings

Proved Cerny conjecture for one-cluster automata

Established quadratic upper bounds for near one-cluster automata

Utilized Markov chains and linear programming in automata theory

Abstract

The \v{C}ern\'y conjecture (\v{C}ern\'y, 1964) states that each n-state \san\ possess a \sw\ of length $(n-1)^2$. From the other side the best upper bound for the \rl\ of n-state \sa\ known so far is equal to $\frac{n^3-n}6$ (Pin, 1983) and so is cubic (a slightly better though still cubic upper bound $\frac{n(7n^2+6n-16)}{48}$ has been claimed in Trahtman but the published proof of this result contains an unclear place) in $n$. In the paper the \v{C}ern\'y conjecture is reduced to a simpler conjecture. In particular, we prove \v{C}ern\'y conjecture for one-cluster automata and quadratic upper bounds for automata closed to one-cluster automata. Our approach utilize theory of Markov chains and one simple fact from linear programming.