Some new Features of Row Monomial Matrices for the Study of DFA

Abstract

The work presents some new results and ideas concerning the synchronization of automata.A word \(W\) of letters on edges of underlying graph \(\Gamma\) of deterministic finite automaton (DFA) is called synchronizing if sends all states of the automaton to a unique state. The problem is decide whether or not deterministic finite automaton (DFA) is synchronizing, find relatively short synchronizing words and such a word of minimal length.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. The hypothesis was formulated in 1966 by Starke.To prove the conjecture, we use algebra of 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 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 \(L_x\).