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

TL;DR

This paper demonstrates that in finite structures, certain classes closed under extensions cannot be defined with limited quantifier alternations, even though they are definable in richer logical frameworks, thus extending known limitations of the preservation theorem.

Contribution

It constructs, for every n, finite classes closed under extensions that are definable but not with n quantifier alternations, answering an open question.

Findings

Constructed classes are definable in Datalog with negation.

Classes are definable in existential transitive-closure logic.

Shows limitations of the finite extension preservation theorem.

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.