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.