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

Anuj Dawar; Abhisekh Sankaran

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arXiv · 2007.05459

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

Anuj Dawar·Abhisekh Sankaran

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Abstract

It is well known that the classic {\L}o\'s-Tarski preservation theorem fails in the finite: there are first-order definable classes of finite structures closed under extensions which are not definable (in the finite) in the existential fragment of first-order logic. We strengthen this by constructing for every $n$, first-order definable classes of finite structures closed under extensions which are not definable with $n$ quantifier alternations. The classes we construct are definable in the extension of Datalog with negation and indeed in the existential fragment of transitive-closure logic. This answers negatively an open question posed by Rosen and Weinstein.

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Anuj Dawar, Abhisekh Sankaran. 2020-07-10. Extension Preservation in the Finite and Prefix Classes of First Order Logic. https://arxiv.org/abs/2007.05459

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