Matrix approach to synchronizing automata

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. Černy 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 Černy 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 distinctly in 1966 by Starke. A special classes of matrices induced by words in the alphabet of labels on edges of the underlying graph of DFA are used for the study of synchronizing automata.

19-pages.3 figures An error removed. arXiv admin note: text overlap with arXiv:1405.2435, arXiv:1202.4626