The length of a minimal synchronizing word and the \v{C}erny conjecture

TL;DR

This paper investigates the Cerny conjecture, proposing an algebraic approach using non-standard matrix operations to analyze the length of minimal synchronizing words in deterministic finite automata.

Contribution

It introduces a novel algebraic method involving non-standard matrix operations to study the length bounds of minimal synchronizing words, aiming to prove the Cerny conjecture.

Findings

Established a connection between word length and matrix space dimension.

Developed an algebraic framework using row monomial matrices.

Provided insights towards proving the Cerny conjecture.

Abstract

A word w of letters on edges of underlying graph Gamma of deterministic finite automaton (DFA) is called the synchronizing word if w sends all states of the automaton to a unique state. J. Cerny 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 the Cerny conjecture, claims that it is also precise upper bound on the length of such a word for a complete DFA. This simple looking conjecture is arguably the most fascinating and longstanding open problem in the theory of finite automaton. The hypothesis was formulated in 1966 by Starke. The problem has motivated great and constantly growing number of investigations and generalizations. To prove the conjecture, we use algebra with non-standard operations on a special class of matrices (row monomial), induced by words in the alphabet of labels on edges. These matrices generate a space with respect to the mentioned operation. The proof is based on connection between length of words u and dimension of the space generated by solutions Lx of matrix equation MuLx =Ms for synchronizing word s, as well as on the relation between ranks of Mu and Lx.