Horn versus full first-order: complexity dichotomies in algebraic constraint satisfaction

TL;DR

This paper establishes complexity dichotomies for infinite-domain constraint satisfaction problems based on algebraic structures, showing they are either in P or NP-hard depending on the relations allowed.

Contribution

It introduces definability dichotomy theorems for algebraic structures, providing a unified framework to classify the complexity of certain infinite-domain CSPs.

Findings

CSPs for expansions of (Q;+) are either in P or NP-hard.

Full dichotomies are obtained for the algebraic and affine variants.

The results unify complexity classification for broad classes of CSPs.

Abstract

We study techniques for deciding the computational complexity of infinite-domain constraint satisfaction problems. For certain fundamental algebraic structures Delta, we prove definability dichotomy theorems of the following form: for every first-order expansion Gamma of Delta, either Gamma has a quantifier-free Horn definition in Delta, or there is an element d of Gamma such that all non-empty relations in Gamma contain a tuple of the form (d,...,d), or all relations with a first-order definition in Delta have a primitive positive definition in Gamma. The results imply that several families of constraint satisfaction problems exhibit a complexity dichotomy: the problems are in P or NP-hard, depending on the choice of the allowed relations. As concrete examples, we investigate fundamental algebraic constraint satisfaction problems. The first class consists of all first-order expansions of (Q;+). The second class is the affine variant of the first class. In both cases, we obtain full dichotomies by utilising our general methods.