Cerny-Starke conjecture from the sixties of XX century

Abstract

A word w of letters on edges of underlying graph Γ 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 Σ of letters on edges of Γ 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 to prove the conjecture.

The algebra of row monomial matrices

“…For every words b and The work ( [29]) used the distributivity only for left multiplier and proves it only for the left factor. As for right multiplier, footnote after proof regarding it is incorrect.…”

“…It is connected, in particular, with the number R(u) of nonzero columns in the matrix of this mapping. The corresponding definition from [29] follows.…”

“…Some matrix M a defines several paths of a in opposite direction from the state q to the set of corresponding states c a . Now follows a key lemma in the proof of the Černy-Starke conjecture [29].…”

“…The problem of road coloring to find a labelling of the edges of a graph that turns the graph into synchronizing DFA was stated in 1970 [1] and solved in 2008 [27], [26]. The proof of Černy's conjecture supposed in [29]. The proof essentially uses the algebra of monomial matrices.…”

Synchronization of finite automata

“…If |Q| = n, then the main loop of Algorithm 1 is executed at most n − 1 times, since after each execution the size of the current set P decreases at least by 1. The 12 From time to time preprints by Trahtman appear that contain arguments which the author proposes as a proof of Černý's conjecture: see [184]- [186]. After a while it turns out that the argument contains some errors and the preprints are withdrawn.…”