Decidability of Definability

For a fixed infinite structure $\Gamma$ with finite signature tau, we study the following computational problem: input are quantifier-free first-order tau-formulas phi_0,phi_1,...,phi_n that define relations R_0,R_1,\dots,R_n over Gamma. The question is whether the relation R_0 is primitive positive definable from R_1,...,R_n, i.e., definable by a first-order formula that uses only relation symbols for R_1, ..., R_n, equality, conjunctions, and existential quantification (disjunction, negation, and universal quantification are forbidden).We show decidability of this problem for all structures Gamma that have a first-order definition in an ordered homogeneous structure Delta with a finite language whose age is a Ramsey class and determined by finitely many forbidden substructures. Examples for structures Gamma with this property are the order of the rationals, the random graph, the homogeneous universal poset, the random tournament, all homogeneous universal C-relations, and many more. We also obtain decidability of the problem when we replace primitive positive definability by existential positive, or existential definability. Our proof makes use of universal algebraic and model theoretic concepts, Ramsey theory, and a recent characterization of Ramsey classes in topological dynamics.