The complexity of positive first-order logic without equality

Madelaine, Florent and Martin, Barnaby (2012) The complexity of positive first-order logic without equality. ACM Transactions on Computational Logic (TOCL), 13 (1) . ISSN 1529-3785 [Article] (doi:10.1145/2071368.2071373)

Abstract

We study the complexity of evaluating positive equality-free sentences of first-order (FO) logic over a fixed, finite structure B. This may be seen as a natural generalisation of the nonuniform quantified constraint satisfaction problem QCSP(B). We introduce surjective hyper-endomorphisms and use them in proving a Galois connection that characterizes definability in positive equality-free FO. Through an algebraic method, we derive a complete complexity classification for our problems as B ranges over structures of size at most three. Specifically, each problem either is in L, is NP-complete, is co-NP-complete, or is Pspace-complete.

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