Alladi Ramakrishnan Hall

A finitary analogue of the downward Lowenheim-Skolem property

Abhisekh Sankaran

IMSc

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 the *equivalent bounded substructure property*, denoted EBSP.
Intuitively, EBSP states that a large finite structure contains a small
``logically similar'' substructure, where logical similarity means
indistinguishability with respect to sentences of FO/MSO 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 regular languages of words, trees (unordered, ordered, ranked or
partially ranked) and nested words, and various classes of graphs, such as
cographs, graph classes of bounded tree-depth, those of bounded shrub-depth
and m-partite cographs. Further, EBSP remains preserved in the classes
generated from the above using various well-studied operations, such as
complementation, transpose, the line-graph operation, disjoint union, join,
and various products including the Cartesian and tensor products.

All of the aforementioned classes admit natural tree representations for
their structures. We use this fact to show that the small and logically
similar substructure of a large structure in any of these classes, can be
computed in time linear in the size of the tree representation of the
structure, giving linear time fixed parameter tractable (f.p.t.) algorithms
for checking FO/MSO definable properties of the large structure. We
conclude by presenting a strengthening of 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.