Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Separating regular languages with two quantifier alternations   We investigate a famous decision problem in automata theory: separation.
Given a class of language C, the separation problem for C takes as input two
regular languages and asks whether there exists a third one which belongs to C,
includes the first one and is disjoint from the second. Typically, obtaining an
algorithm for separation yields a deep understanding of the investigated class
C. This explains why a lot of effort has been devoted to finding algorithms for
the most prominent classes.
  Here, we are interested in classes within concatenation hierarchies. Such
hierarchies are built using a generic construction process: one starts from an
initial class called the basis and builds new levels by applying generic
operations. The most famous one, the dot-depth hierarchy of Brzozowski and
Cohen, classifies the languages definable in first-order logic. Moreover, it
was shown by Thomas that it corresponds to the quantifier alternation hierarchy
of first-order logic: each level in the dot-depth corresponds to the languages
that can be defined with a prescribed number of quantifier blocks. Finding
separation algorithms for all levels in this hierarchy is among the most famous
open problems in automata theory.
  Our main theorem is generic: we show that separation is decidable for the
level 3/2 of any concatenation hierarchy whose basis is finite. Furthermore, in
the special case of the dot-depth, we push this result to the level 5/2. In
logical terms, this solves separation for $\Sigma_3$: first-order sentences
having at most three quantifier blocks starting with an existential one.

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The categories of this paper are: atom-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: dg-ga
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The categories of this paper are: eess
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The categories of this paper are: plasm-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-th
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The categories of this paper are: math-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: quant-ph
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The categories of this paper are: math
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The categories of this paper are: econ
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The categories of this paper are: adap-org
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The categories of this paper are: chao-dyn
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The categories of this paper are: cmp-lg