The \v{C}erny conjecture

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 the \v{C}erny conjecture, claims that it 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. To prove the conjecture, we use algebra w on a special class of row monomial matrices (one unit and rest zeros in every row), 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 $L_x$ of matrix equation $M_uL_x=M_s$ for synchronizing word $s$, as well as on the relation between ranks of $M_u$ and $L_x$.