Optimization of Disjunctive Abstract Interpretations

We define the inverse operation for disjunctive completion, introducing the notion of least disjunctive basis for 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 in relation with reduced product of abstract interpretations. The resulting framework is powerful enough to be applied to arbitrary abstract domains for analysis, providing advanced algebraic methods for domain manipulation and optimization. These notions are applied to abstract domains for analysis of functional and logic programming languages.