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. \v{C}erny 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 \v{C}erny 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 to prove the conjecture. The last one is based on connection between length of $u$ and dimension of the space generated by solution $L_x$ of matrix equation $M_uL_x=M_s$ for synchronizing word $s$.

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二十世纪六十年代的塞尔尼-斯塔克猜想

确定性有限自动机 (DFA) 的底层图 $\Gamma$ 边上的字母单词 $w$ 被称为同步，如果 $w$ 将自动机的所有状态发送到唯一状态。J. \v{C}erny 在 1964 年发现了一个 $n$ 状态的完整 DFA 序列，它拥有一个长度为 $(n-1)^2$ 的最小同步字。这个假设，今天众所周知的 \v{C}erny 猜想，声称 $(n-1)^2$ 是这样一个单词在字母表 $\Sigma$ 边缘上的字母长度的精确上限\Gamma$ 对于每个完整的 $n$ 状态 DFA。这个假设是由斯塔克在 1966 年明确提出的。由 DFA 底层图的边上的标签字母表中的单词诱导的一类特殊矩阵用于证明该猜想。