The natural duality between logic and computation becomes clear when the higher order operations of the lambda calculus are combined with Horn clause logic programming. The result is approximately the Tarski-Givant set theory. In it a computational proof of the validity of a relational expression corresponds to a search for a finite representation in set theory. In particular, relation complement is a mapping between finite equational theories in which the operational and denotational semantics agree. These notions are natural consequences of program structuring objectives.
Mathematical notions have intuitive computational interpretations.
Program optimization is justified by a proof of relation equivalence. Program execution corresponds to a proof of a representation.