  For a fixed countably infinite structure \Gamma\ with finite relational
signature \tau, we study the following computational problem: input are
quantifier-free \tau-formulas \phi_0,\phi_1,...,\phi_n that define relations
R_0,R_1,...,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 relational signature whose age is a Ramsey class and determined by
finitely many forbidden substructures. Examples of 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.
