Local higher-order fixpoint iteration

Abstract

Local fixpoint iteration describes a technique that restricts fixpoint iteration in function spaces to needed arguments only. It has been studied well for first-order functions in abstract interpretation and also in model checking. Here we consider the problem for least and greatest fixpoints of arbitrary type order. We define an abstract algebra of simply-typed higher-order functions with fixpoints in order to express fixpoint evaluation problems as they occur routinely in various applications, including program verification. We present an algorithm that realises local fixpoint iteration for such higher-order fixpoints, prove its correctness and study its optimisation potential in the context of several applications. We also examine a particular fragment of this higher-order fixpoint algebra which allows us to pre-compute needed arguments, as this may help to speed up the fixpoint iteration process.