Complexity of Existential Positive First-Order Logic

TL;DR

This paper investigates the computational complexity of deciding existential positive first-order logic sentences over structures, establishing conditions for Logspace decidability or NP-completeness related to the structure's CSP.

Contribution

It characterizes the complexity of the problem, showing it is either in Logspace or NP-complete, depending on the structure, and relates it to the class CSP(gamma)_NP.

Findings

Deciding existential positive sentences is in Logspace or NP-complete.

The complexity depends on properties of the structure gamma.

The problem reduces to CSP(gamma)_NP under polynomial-time many-one reductions.

Abstract

Let gamma be a (not necessarily finite) structure with a finite relational signature. We prove that deciding whether a given existential positive sentence holds in gamma is in Logspace or complete for the class CSP(gamma)_NP under deterministic polynomial-time many-one reductions. Here, CSP(gamma)_NP is the class of problems that can be reduced to the Constraint Satisfaction Problem of gamma under non-deterministic polynomial-time many-one reductions.