A Finitary Analogue of the Downward Löwenheim-Skolem Property

Abstract

We present a model-theoretic property of finite structures, that can be seen to be a finitary analogue of the well-studied downward Löwenheim-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.