Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Regular Separability and Intersection Emptiness are Independent Problems   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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The categories of this paper are: chao-dyn
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The categories of this paper are: econ
10
The categories of this paper are: q-bio
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The categories of this paper are: adap-org
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The categories of this paper are: patt-sol
12
The categories of this paper are: nucl-th
37
The categories of this paper are: supr-con
2
The categories of this paper are: math
35
The categories of this paper are: mtrl-th
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The categories of this paper are: hep-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: funct-an
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The categories of this paper are: atom-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: chem-ph
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The categories of this paper are: solv-int