Synchronization of Some DFA

Trahtman, A. N.

doi:10.1007/978-3-540-72504-6_21

“…An attempt to prove this hypothesis is proposed below. This famous conjecture is true for a lot of automata, but in general the problem still remains open although several hundreds of articles consider this problem from different points of view [7], [12], [17].…”

Section: Introductionmentioning

confidence: 99%

Row monomial matrices and Černy conjecture, short proof

Trahtman¹

2022

Preprint

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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 Γ of deterministic finite automaton (DFA) is called synchronizing if w sends all states of the automaton to a unique state (|R(w)| = 1. 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 the Černy conjecture, claims that (n − 1) 2 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. The proof of the conjecture is based on connection between length of words u and dimension of the space generated by row monomial matrices Mu, the set of synchronizing matrices placed some role.

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“…The package TESTAS [38], [44] studied all automata with strongly connected underlying graph of size n ≤ 11 for |Σ| = 2, of size n ≤ 8 for |Σ| ≤ 3 and of size n ≤ 7 for |Σ| ≤ 4 and found five new examples of DFA with shortest synchronizing word of length (n − 1) 2 for n ≤ 4.…”

Section: Introductionmentioning

confidence: 99%

Cerny-Starke conjecture from the sixties of XX century

Trahtman

2020

Preprint

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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.

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“…The package TESTAS [37], [43] studied all automata with strongly connected underlying graph of size n ≤ 11 for |Σ| = 2, of size n ≤ 8 for |Σ| ≤ 3 and of size n ≤ 7 for |Σ| ≤ 4 and found five new examples of DFA with shortest synchronizing word of length (n − 1) 2 for n ≤ 4.…”

Section: Introductionmentioning

confidence: 99%

Matrix approach to synchronizing automata

Trahtman

2019

Preprint

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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 for the study of synchronizing automata.

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Synchronization of Some DFA

Cited by 4 publications

References 10 publications

Row monomial matrices and Černy conjecture, short proof

Row monomial matrices and Černy conjecture, short proof

Cerny-Starke conjecture from the sixties of XX century

Matrix approach to synchronizing automata

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