Compositional 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. 1 I n t r o d u c t i o n It is widely acknowledged that most program properties need relational abstract domains to be attacked by abstract interpretation ([18, 24]). The Cousot and Cousot functional combination by reduced power ([S]), and Nielson's tensor product ([25]) were the first systematic methods to induce relational analyses by combining abstract domains. Cousot and Cousot showed in [9] that a relational analysis can be induced by combining reduced product (denoted n) and disjunctive completion (denoted G) of abstract domains. If D1 and D~ are abstract domains, a corresponding domain for relational analysis can always be defined as ~3(D1 I-1 D~). In this construction, reduced product is attribute independent (viz. the information obtainable from the combination of analyses is essentially the same as the one obtainable by performing the analyses separately), while disjunctive completion introduces relational information by exploiting sets of attribute independent abstract properties. Disjunctive completion is therefore fundamental to implement relational analyses. Disjunctive completion was originally introduced to exploit disjunctive program properties, notably to prove that merge.over-all-paths (MOP) data-flow analysis can be always expressed in fixpoint form ([8]). This notion was also considered in Nielson's approach to abstract interpretation using domain theory ([24]), and applied in data-flow analysis of functional and logic languages, e.g., to express disjunctive information in Jensen's strictness logic ([17]), in Cousot and Cousot comportment analysis ([10]), and in analysis of ground-dependencies ([12]). A natural question is: can we invert a process of "domain refinement"? Namely, can we reconstruct the "least basis" which induces a given domain by composition