A Generalization of the Łoś-Tarski Preservation Theorem [article]

Abhisekh Sankaran
2016 arXiv   pre-print
In this dissertation, we present for each natural number k, semantic characterizations of the ∃^k ∀^* and ∀^k ∃^* prefix classes of first order logic sentences, over all structures finite and infinite. This result, that we call the *generalized Łoś-Tarski theorem*, abbreviated GLT(k), yields the classical Łoś-Tarski preservation theorem when k equals 0. It also provides new characterizations of the Σ^0_2 and Π^0_2 prefix classes, that are finer than all characterizations of these classes in the
more » ... literature. Further, our semantic notions are finitary in nature, in contrast to those contained in the literature characterizations. In the context of finite structures, we formulate an abstract combinatorial property of structures, that when satisfied by a class, ensures that GLT(k) holds over the class. This property, that we call the *Equivalent Bounded Substructure Property*, abbreviated EBSP, intuitively states that a large structure contains a small "logically similar" substructure. It turns out that this simply stated property is enjoyed by a variety of classes of interest in computer science: examples include words, trees (unordered, ordered or ranked), nested words, graph classes of bounded tree-depth/shrub-depth, and m-partite cographs. Further, EBSP remains preserved under various well-studied operations, such as complementation, transpose, the line-graph operation, disjoint union, cartesian and tensor products, etc. This enables constructing a wide spectrum of classes that satisfy EBSP, and hence GLT(k). Remarkably, EBSP can be regarded as a finitary analogue of the classical downward Löwenheim-Skolem property. In summary, this dissertation provides new notions and results in both contexts, that of all structures and that of finite structures.
arXiv:1609.06297v1 fatcat:6kucahkypzet3mm3kxpwjhu7py