A Finitary Analogue of the Downward Löwenheim-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 Löwenheim-Skolem property from classical model theory. We call this property the L-equivalent bounded substructure property, denoted L-EBSP, where L is either FO or MSO. Intuitively, L-EBSP states that a large finite structure contains a small “logically similar” substructure, where logical similarity means indistinguishability with respect to sentences of 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 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, L-EBSP remains preserved in the classes generated from the above by operations that are implementable using quantifier-free translation schemes. 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 L-definable properties of the large structure. We conclude by presenting a strengthening of L-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. 1998 ACM Subject Classification F.4.1 Model theory, F.4.3 Formal Languages, G.2.2 Graph theory