Complementation in abstract interpretation

Reduced product of abstract domains is a rather well-known operation for domain composition in abstract interpretation. In this article, we study its inverse operation, introducing a notion of domain complementation in abstract interpretation. Complementation provides a systematic way to design new abstract domains, and it allows to systematically decompose domains. Also, such an operation allows to simplify domain verification problems, and it yields space-saving representations for complex domains. We show that the complement exists in most cases, and we apply complementation to three well-known abstract domains, notably to Cousot and Cousot's interval domain for integer variable analysis, to Cousot and Cousot's domain for comportment analysis of functional languages, and to the domain Sharing for aliasing analysis of logic languages.