关于有序DFA的一点注记
A Note on Ordinal DFAs
摘要：
我们证明了下面的定理。假设$M$是布尔字母表$0,1$上的修剪DFA。语言$Ł(M)$是按字典序$\slex$良序的当且仅当非汇状态$q,q.0$在同一个强分量中时，则$q.1$是汇。很容易看出，这个属性是足够的。为了说明这种必要性，我们分析了一个$\slex$ -降序词序列的行为。该属性用于获得多项式时间算法，以在给定DFA $M$的情况下确定$Ł(M)$是否按字典顺序良好排序。最后，我们应用\cite{BE,BEa}中的一个论证证明了最小非正则序数是$ω^ω$ .
英文摘要：
We prove the following theorem. Suppose that $M$ is a trim DFA on the Boolean alphabet $0,1$. The language $Ł(M)$ is well-ordered by the lexicographic order $\slex$ iff whenever the non sink states $q,q.0$ are in the same strong component, then $q.1$ is a sink. It is easy to see that this property is sufficient. In order to show the necessity, we analyze the behavior of a $\slex$-descending sequence of words. This property is used to obtain a polynomial time algorithm to determine, given a DFA $M$, whether $Ł(M)$ is well-ordered by the lexicographic order. Last, we apply an argument in \cite{BE,BEa} to give a proof that the least nonregular ordinal is $ω^ω$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
