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Revisiting the Generalized Łoś-Tarski Theorem

  • Abhisekh SankaranEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11600)

Abstract

We present a new proof of the generalized Łoś-Tarski theorem (\(\mathsf {GLT}({k})\)) from [6], over arbitrary structures. Instead of using \(\lambda \)-saturation as in [6], we construct just the “required saturation” directly using ascending chains of structures. We also strengthen the failure of \(\mathsf {GLT}({k})\) in the finite shown in [7], by strengthening the failure of the Łoś-Tarski theorem in this context. In particular, we prove that not just universal sentences, but for each fixed k, even \(\varSigma ^0_2\) sentences containing k existential quantifiers fail to capture hereditariness in the finite. We conclude with two problems as future directions, concerning the Łoś-Tarski theorem and \(\mathsf {GLT}({k})\), both in the context of all finite structures.

Keywords

Łoś-Tarski theorem k-hereditary k-ary cover Chain 

Notes

Acknowledgments

I would like to thank Anuj Dawar for pointing out the Ehrenfeucht-Fräissé game perspective to the arguments contained in the proof of Theorem 4. I also thank the anonymous referees for their comments and suggestions.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Computer Science and TechnologyUniversity of CambridgeCambridgeUK

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