A Generalization of the Łoś-Tarski Preservation Theorem over Classes of Finite Structures

Abstract

We investigate a generalization of the Loś-Tarski preservation theorem via the semantic notion of preservation under substructures modulo k-sized cores. It was shown earlier that over arbitrary structures, this semantic notion for first-order logic corresponds to definability by ∃ k ∀ * sentences. In this paper, we identify two properties of classes of finite structures that ensure the above correspondence. The first is based on well-quasi-ordering under the embedding relation. The second is a logic-based combinatorial property that strictly generalizes the first. We show that starting with classes satisfying any of these properties, the classes obtained by applying operations like disjoint union, cartesian and tensor products, or by forming words and trees over the classes, inherit the same property. As a fallout, we obtain interesting classes of structures over which an effective version of the Loś-Tarski theorem holds.

A generalization of the Łoś–Tarski preservation theorem

“…The latter section also contains our answer to the second part of Q2 raised above. The results that we present here contain, and generalize significantly, the results in [73,75,76]. …”

“…Techniques: We prove an abstract result concerning tree representations (Theorem 10.1.1), that takes as input a tree-representation of a structure and produces as output, a small subtree that represents a small and logically similar substructure of the original structure. The output structure is obtained by iteratively performing appropriate "prunings" of, and "graft- In connection with my published papers, I thank the anonymous reviewers of [73,75,76] for their careful reading of the earlier versions of these, and for their many keen suggestions on improving the presentation of the results therein. I also thank Pritish Kamath and Vivek…”