Online Appendix to The Recovery of a Schema Mapping: Bringing Exchanged Data Back

To prove the lemma, we provide an algorithm that given an st-mapping M = (S,T,Σ) such that Σ is a set of FO-TO-CQ dependencies, and a conjunctive query Q over schema T, computes a queryQ that is a rewriting of Q over the source schema S. We first introduce the terminology used in the algorithm. The basic notion used in the algorithm is that of existential replacement. In an existential replacement of a formula β, we are allowed to existentially quantify some of the positions of the free variables of β. For example, if β(x1, x2, x3) = P (x1, x2) ∧ R(x2, x3), then two existential replacements of β(x1, x2, x3) are γ1(x2) = ∃u∃v (P (u, x2) ∧ R(x2, v)) and γ2(x1, x2, x3) = ∃z (P (x1, z) ∧ R(x2, x3)). We note that both γ1 and γ2 are implied by β. In an existential replacement, we are also allowed to use the same quantifier for different positions. For example, γ3(x2) = ∃w (P (w, x2) ∧ R(x2, w)) is also an existential replacement of β. We note that γ3 is implied by β if x1 and x3 have the same value, that is, β(x1, x2, x3) ∧ x1 = x3 implies γ3. In an existential replacement, these equalities are also included. Formally, given a formula β(x̄) where x̄ = (x1, . . . , xk) is a tuple of distinct variables, an existential replacement of β(x̄) is a pair of formulas (∃z̄ γ(x̄, z̄), θ(x̄)), where: (1) ∃z̄ γ(x̄, z̄) is obtained from β(x̄) by existentially quantifying some of the positions of the free variables of β(x̄), and z̄ is the tuple of fresh variables used in these quantifications, (2) θ(x̄) is a conjunction of equalities such that xi = xj is in θ (1 ≤ i, j ≤ k and i 6= j) if we replace a position with variable xi and a position with variable xj by the same variable z from z̄, and (3) x̄ ′ and x̄ are the tuples of free variables of ∃z̄ γ(x̄, z̄) and θ(x̄), respectively. Notice that ∃z̄ γ(x̄, z̄) is a logical consequence