Revisiting the Generalized Łoś-Tarski Theorem

Abstract

We present a new proof of the generalized Loś-Tarski theorem (GLT(k)) introduced in [6], over arbitrary structures. Instead of using λ-saturation as in [6], we construct just the "required saturation" directly using ascending chains of structures. We also strengthen the failure of GLT(k) in the finite shown in [7], by strengthening the failure of the Loś-Tarski theorem in this context. In particular, we prove that not just universal sentences, but for each fixed k, even Σ 0 2 sentences containing k existential quantifiers fail to capture hereditariness in the finite. We conclude with two problems as future directions, concerning the Loś-Tarski theorem and GLT(k), both in the context of all finite structures.1 See [5] for a variety of graph properties of interest in parameterized algorithms and finite model theory, that are k-hereditary and expressible as ∃ k ∀ * sentences.

Extension Preservation in the Finite and Prefix Classes of First Order Logic

“…Our result also greatly strengthens a previous result by Sankaran [15] which showed that for each k there is an extension-closed property of finite structures definable in FO but not in Π 2 with k leading universal quantifiers. Indeed, our result answers (negatively) Problem 2 in [15].…”

Forbidden Induced Subgraphs and the Łoś–tarski Theorem

“…Acknowledgment. We thank Abhisekh Sankaran for mentioning to the first author the question of whether Tait's Theorem generalizes to graphs (see also [23]).…”

“…Let f : N → N be a computable function. Furthermore, let w be defined by (23), where I is an interpretation for := M according to Lemma 5.6. Then there is a w ∈ {0, 1} * such that Graph( w ) is closed under induced subgraphs (and hence equivalent in the class of graphs to a universal sentence) but w is not equivalent in the class of finite graphs to a universal sentence of length less than f(| w |).…”