Combinations of abstract domains for logic programming: open product and generic pattern construction

Abstract interpretation is a systematic methodology to design static program analysis which has been studied extensively in the logic programming community, because of the potential for optimizations in logic programming compilers and the sophistication of the analyses which require conceptual support. With the emergence of efficient generic abstract interpretation algorithms for logic programming, the main burden in building an analysis is the abstract domain which gives a safe approximation of the concrete domain of computation. However, accurate abstract domains for logic programming are often complex not only because of the relational nature of logic programming languages and of their typical interprocedural control-flow, but also because of the variety of analyses to perform, their interdependence, and the need to maintain structural information. The purpose of this paper is to propose conceptual and software support for the design of abstract domains. It contains two main contributions: the notion of open product and a generic pattern domain. The open product is a new, language independent, way of combining abstract domains allowing each combined domain to benefit from information from the other components through the notions of queries and open operations. It provides a framework to approximate Cousots’ reduced product, while reusing existing implementations and providing methodological guidance on how to build domains for interaction and composition. It is orthogonal and complementary to Granger's product which improves the direct product by a decreasing iteration sequence based on refinements but lets the domains interact only after the individual operations. The generic pattern domain Pat( R ) automatically upgrades a domain D with structural information yielding a more accurate domain Pat(D) without additional design or implementation cost. The two contributions are orthogonal and can be combined in various ways to obtain sophisticated domains while imposing minimal requirements on the designer. Both contributions are characterized theoretically and experimentally and were used to design very complex abstract domains such as PAT(OPos⊗OMode⊗OPS) which would be very difficult to design otherwise. On this last domain, designers need only contribute about 20% (about 3400 lines) of the complete system (about 17,700 lines).