A logical model for relational abstract domains

In this article we introduce the notion of Heyting completion in abstract interpretation. We prove that Heyting completion provides a model for Cousot's reduced cardinal power of abstract domains and that it supplies a logical basis to specify relational domains for program analysis and abstract interpretation. We study the algebraic properties of Heyting completion in relation with other well-known domain transformers, like reduced product and disjunctive completion. This provides a uniform algebraic setting where complex abstract domains can be specified by simple logic formulas, or as solutions of recursive abstract domain equations, involving few basic operations for domain construction, all characterized by a clean logical interpretation. We apply our framework to characterize directionality and condensing in downward closed analysis of (constraint) logic programs.