Cerny-Starke conjecture from the sixties of XX century

Abstract

A word $s$ of letters on edges of underlying graph $\Gamma$ of deterministic finite automaton (DFA) is called synchronizing if $s$ 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, mostly known today as \v{C}erny conjecture, claims that $(n-1)^2$ is a precise upper bound on the length of such a word over alphabet $\Sigma$ of letters on edges of $\Gamma$ for every complete $n$-state DFA. The hypothesis was formulated in 1966 by Starke. Algebra with nonstandard operation over special class of matrices induced by words in the alphabet of labels on edges is used to prove the conjecture. The proof is based on the connection between length of words $u$ and dimension of the space generated by solution $L_x$ of matrix equation $M_uL_x=M_s$ for synchronizing word $s$, as well as on relation between ranks of $M_u$ and $L_x$. Important role below placed the notion of pseudo inverseL matrix, sometimes reversible.