Query reformulation with constraints

Let Σ1, Σ2 be two schemas, which may overlap, C be a set of constraints on the joint schema Σ1 ∪ Σ2, and q1 be a Σ1-query. An (equivalent) reformulation of q1 in the presence of C is a Σ2-query, q2, such that q2 gives the same answers as q1 on any Σ1 ∪ Σ2-database instance that satisfies C. In general, there may exist multiple such reformulations and choosing among them may require, for example, a cost model