The reduced relative power operation on abstract domains

In the context of standard abstract interpretation theory, a reduced relative power operation for functionally composing abstract domains is introduced and studied. The reduced relative power of two abstract domains D-1 (the exponent) and D-2 (the base) consists in a suitably defined lattice of monotone functions from D-1 to D-2, called dependencies, and it is a generalization of the Cousot and Cousot reduced cardinal power operation. The relationship between reduced relative power and Nielson's tensor product of abstract domains is also investigated. The case of autodependencies, i.e, base and exponent are the same domain, turns out to be particularly interesting: Under certain hypotheses, the domain of autodependencies corresponds to a powerset-like completion of the base abstract domain, providing a compact set-theoretic representation for autodependencies. Two relevant applications of the reduced relative power operation in the fields of logic program analysis and semantics design are presented. Notably, it is proved that the well-known abstract domain Def for logic program ground-dependency analysis can be characterized as the domain of autodependencies of the standard abstract domain representing plain groundness information only; on the semantics side, it is shown how reduced relative power can be exploited in order to systematically derive compositional semantics for logic programs.