Higher-order abstract interpretation (and application to comportment analysis generalizing strictness, termination, projection and PER analysis of functional languages)

The original formulation of abstract interpretation represents program properties by sets. A property is understood as the set of semantic values satisfying it. Strongest program properties are defined by the collecting semantics which extends the standard semantics to powersets of semantic values. The approximation relation corresponding to the logical implication of program properties is subset inclusion. This was expressed using set and lattice theory in the context of transition systems. Some applications of abstract interpretation, such as strictness analysis for lazy functional languages, require infinite behaviours of higher-order functions to be taken into account. We solve the problem by returning to the sources of abstract interpretation, which consists in considering collecting semantics. By using Galois connections, properties of the standard semantics naturally transfer to the collecting and then to the abstract semantics. This set-theoretic abstract interpretation framework is formulated in a way which is independent of both the programming language and the method used to specify its semantics. It is illustrated for a higher-order monomorphically typed lazy functional language starting from its standard denotational semantics