Row monomial matrices and \v{C}erny conjecture, short proof

TL;DR

This paper proves the Cerny conjecture, establishing that the shortest synchronizing word in a complete DFA with n states is at most (n-1)^2, using properties of row monomial matrices and graph path analysis.

Contribution

The paper provides a short proof of the Cerny conjecture, connecting matrix space dimensions with automaton synchronizing word lengths.

Findings

Proved the Cerny conjecture for all complete DFA.

Established a bound of (n-1)^2 on synchronizing word length.

Linked matrix space properties to automaton synchronization.

Abstract

The class of row monomial matrices (one unit and rest zeros in every row) with some non-standard operations of summation and usual multiplication is our main object. These matrices generate a space with respect to the mentioned operations. A word w of letters on edges of underlying graph 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)(n-1). The hypothesis, well known today as the \v{C}erny conjecture, claims that (n-1)(n-1) is also precise upper bound on the length of such a word for a complete DFA. The hypothesis was formulated in 1966 by Starke. The problem has motivated great and constantly growing number of investigations and generalizations. We present the proof of the \v{C}erny-Starke conjecture: the deterministic complete n-state synchronizing automaton has synchronizing word of length at most (n-1)(n-1). The proof used connection between dimension of the space and the length of words on paths of edges in underlying graph of automaton.