正则可分性和交空性是独立的问题
Regular Separability and Intersection Emptiness are Independent Problems
摘要：
\emph{regular separability}的问题是，给定两种语言$K$和$L$，是否存在具有$K\subseteq S$和$S\cap L=\emptyset$的正则语言$S$。这个问题最近已经研究了各种类型的语言。到目前为止，所有关于正则可分性的结果都与交集空性问题表现出了值得注意的对应关系：在每种情况下，正则可分性是可判定的当且仅当交集空性是可判定的。这就提出了一个问题，在温和的假设下，正则可分性是否可以减少到交集空，反之亦然。我们提出的反例表明，这两个问题都不能减少到其他。更具体地说，我们描述了语言类$\mathcal{C_1}$、$\mathcal{D_1}$、$\mathcal{C_2}$、使得（i）对于$\mathcal{C_1}$和$\mathcal{D_1}$，但正则可分性对$\mathcal{C_1}$和$\mathcal{D_1}$是不可判定的;（ii）正则可分性是可判定的可用于$\mathcal{C_2}$和$\mathcal{D_2}$，但对于$\mathcal{C_2}$和$\mathcal{D_2}$。
英文摘要：
The problem of \emph{regular separability} asks, given two languages $K$ and $L$, whether there exists a regular language $S$ with $K\subseteq S$ and $S\cap L=\emptyset$. This problem has recently been studied for various classes of languages. All the results on regular separability obtained so far exhibited a noteworthy correspondence with the intersection emptiness problem: In eachcase, regular separability is decidable if and only if intersection emptiness is decidable. This raises the question whether under mild assumptions, regular separability can be reduced to intersection emptiness and vice-versa. We present counterexamples showing that none of the two problems can be reduced to the other. More specifically, we describe language classes $\mathcal{C_1}$, $\mathcal{D_1}$, $\mathcal{C_2}$, $\mathcal{D_2}$ such that (i)~intersection emptiness is decidable for $\mathcal{C_1}$ and $\mathcal{D_1}$, but regular separability is undecidable for $\mathcal{C_1}$ and $\mathcal{D_1}$ and (ii)~regular separability is decidable for $\mathcal{C_2}$ and $\mathcal{D_2}$, but intersection emptiness is undecidable for $\mathcal{C_2}$ and $\mathcal{D_2}$.
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