2022/03/28 by A. N. Trahtman, Trahtman, A. N. · 1 citation
Computer Science · Mathematics · #05C50 #05C85 #68R10 #Advanced Combinatorial Mathematics #Discrete Mathematics (cs.DM) #F.2.2 #FOS: Computer and information sciences #Formal Languages and Automata Theory (cs.FL) #Graph theory and applications #I.2.7 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2203.14822
openalex publication_date 2022/03/28 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28
The class of row monomial matrices (one unit and rest zeros in every row) with some non-standard operations of summation and usual multiplication is our main object. These matrices generate a space with respect to the mentioned operations. A word w of letters on edges of underlying graph 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)(n-1). The hypothesis, well known today as the Černy conjecture, claims that (n-1)(n-1) 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. We present the proof of the Černy-Starke conjecture: the deterministic complete n-state synchronizing automaton has synchronizing word of length at most (n-1)(n-1). The proof used connection between dimension of the space and the length of words on paths of edges in underlying graph of automaton.