Generalizations of the Los-Tarski Preservation Theorem

TL;DR

This paper introduces new semantic preservation theorems for specific quantifier prefix classes in first-order logic, generalizing classical results and providing insights into model properties and theories.

Contribution

It establishes novel preservation theorems relating quantifier counts to model properties, extending the classical Los-Tarski theorem to broader logical classes and theories.

Findings

Generalized Los-Tarski preservation theorem for finite vocabularies

Semantic characterization of $orall^k orall^*$ and $orall^k orall^*$ classes

Interpolant-based approach to preservation theorems

Abstract

We present new preservation theorems that semantically characterize the $\exists^k \forall^*$ and $\forall^k \exists^*$ prefix classes of first order logic, for each natural number $k$. Unlike preservation theorems in the literature that characterize the $\exists^* \forall^*$ and $\forall^* \exists^*$ prefix classes, our theorems relate the count of quantifiers in the leading block of the quantifier prefix to natural quantitative properties of the models. As special cases of our results, we obtain the classical Los-Tarski preservation theorem for sentences in both its extensional and substructural versions. For arbitrary finite vocabularies, we also generalize the extensional version of the Los-Tarski preservation theorem for theories. We also present an interpolant-based approach towards these results. Finally, we present partial results towards generalizing to theories, the substructural version of the Los-Tarski theorem and in the process, we give a preservation theorem that provides a semantic characterization of $\Sigma^0_n$ theories for each natural number $n$.