Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Finitary Analogue of the Downward L\"owenheim-Skolem Property   We present a model-theoretic property of finite structures, that can be seen
to be a finitary analogue of the well-studied downward L\"owenheim-Skolem
property from classical model theory. We call this property as the
*$\mathcal{L}$-equivalent bounded substructure property*, denoted
$\mathcal{L}$-$\mathsf{EBSP}$, where $\mathcal{L}$ is either FO or MSO.
Intuitively $\mathcal{L}$-$\mathsf{EBSP}$ states that a large finite structure
contains a small "logically similar" substructure, where logical similarity
means indistinguishability with respect to sentences of $\mathcal{L}$ having a
given quantifier nesting depth. It turns out that this simply stated property
is enjoyed by a variety of classes of interest in computer science: examples
include various classes of posets, such as regular languages of words, trees
(unordered, ordered or ranked) and nested words, and various classes of graphs,
such as cographs, graph classes of bounded tree-depth, those of bounded
shrub-depth and $n$-partite cographs. Further, $\mathcal{L}$-$\mathsf{EBSP}$
remains preserved in the classes generated from the above by operations that
are implementable using quantifier-free translation schemes. We show that for
natural tree representations for structures that all the aforementioned classes
admit, the small and logically similar substructure of a large structure can be
computed in time linear in the size of the representation, giving linear time
fixed parameter tractable (f.p.t.) algorithms for checking $\mathcal{L}$
definable properties of the large structure. We conclude by presenting a
strengthening of $\mathcal{L}$-$\mathsf{EBSP}$, that asserts "logical
self-similarity at all scales" for a suitable notion of scale. We call this the
*logical fractal* property and show that most of the classes mentioned above
are indeed, logical fractals.

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The categories of this paper are: comp-gas
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The categories of this paper are: dg-ga
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The categories of this paper are: astro-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: stat
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The categories of this paper are: q-alg
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-fin
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The categories of this paper are: solv-int
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The categories of this paper are: chem-ph
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The categories of this paper are: physics
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The categories of this paper are: gr-qc