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

Abstract

We study techniques for deciding the computational complexity of infinite-domain constraint satisfaction problems. For certain basic algebraic structures Δ, we prove definability theorems of the following form: for every first-order expansion Γ of Δ, either Γ has a quantifier-free Horn definition in Δ, or there is an element d of Γ such that all non-empty relations in Γ contain a tuple of the form (d,…,d), or all relations with a first-order definition in Δ have a primitive positive definition in Γ. The results imply that several families of constraint satisfaction problems exhibit a complexity dichotomy: the problems are either polynomial-time solvable 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 relational structures with a first-order definition in (ℚ; +) that contain the relation {(x, y, z) ∈ ℚ3 | x + y = z}. The second class is the affine variant of the first class. In both cases, we obtain full dichotomies by utilizing our general methods.