Optimal domains for disjunctive abstract interpretation

In the context of standard abstract interpretation theory, we define the inverse operation to the disjunctive completion of abstract domains, introducing the notion of least disjunctive basis of an abstract domain D. This is the most abstract domain inducing the same disjunctive completion as D. We show that the least disjunctive basis exists in most cases, and study its properties, also in relation with reduced product and complementation of abstract domains. The resulting framework is powerful enough to be applied to arbitrary abstract domains for analysis, providing advanced algebraic methodologies for domain manipulation and optimization. These notions are applied to abstract domains for static analysis of functional and logic programming languages.