A word $w$ is called a reaching word of a subset $S$ of states in a deterministic finite automaton (DFA) if $S$ is the image of $Q$ under the action of $w$. A DFA is called completely reachable if every non-empty subset of the state set has a reaching word. A conjecture states that in every $n$-state completely reachable DFA, for every $k$-element subset of states, there exists a reaching word of length at most $n(n-k)$. We present infinitely many completely reachable DFAs with two letters that violate this conjecture. A subfamily of completely reachable DFAs with two letters, is called standardized DFAs, introduced by Casas and Volkov (2023). We prove that every $k$-element subset of states in an $n$-state standardized DFA has a reaching word of length $\le n(n-k) + n - 1$. Finally, we confirm the conjecture for standardized DFAs with additional properties, thus generalizing a result of Casas and Volkov (2023).

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围绕Don关于二元完全可达自动机的猜想

如果 $S$ 是 $Q$ 在 $w$ 作用下的图像，则单词 $w$ 称为确定性有限自动机 (DFA) 中状态子集 $S$ 的到达词。如果状态集的每个非空子集都有到达字，则 DFA 称为完全可达。一个猜想指出，在每个 $n$ 状态完全可达 DFA 中，对于状态的每个 $k$ 元素子集，都存在长度至多 $n(nk)$ 的可达字。我们提出了无限多个完全可达的 DFA，其中两个字母违反了这个猜想。具有两个字母的完全可达 DFA 的子族称为标准化 DFA，由 Casas 和 Volkov (2023) 提出。我们证明，$n$ 状态标准化 DFA 中的每个 $k$ 元素子集都有一个长度为 $\le n(nk) + n - 1$ 的到达词。最后，我们证实了具有附加属性的标准化 DFA 的猜想，从而概括了 Casas 和 Volkov (2023) 的结果。