BP-Completeness

DEFINITION A relational query language Q is BP-complete if for each relational database D, the set of all relations defined by the queries of Q on D is equal to the set of all first-order definable relations over D. More formally, fix some infinite universe U of atomic data elements. A relational database schema S is a finite set of relation names, each with an associated arity. A relational database D with schema S assigns to each relation name of S a finite relation over U of its arity. The domain of D, dom(D), is the set of all atomic data elements occurring in the tuples of its relations. Let FO be the set of first-order formulas over signature S and the equality predicate, and let FO(D) = {φ(D) | φ ∈ FO}. (For a formula φ ∈ FO with free variables (x1, . . . , xm), φ(D) denotes the m-ary relation over dom(D) defined by φ, where the variables in φ are assumed to range over dom(D).) Let Q denote those queries of Q defined over schema S , and let Q(D) = {q(D) | q ∈ Q}, i.e., the set of relations defined by queries of Q applied to D. Then, Q is BPcomplete if for each relational database D over schema S,