This paper gives the first complete axiomatization for higher types in the refinement calculus of predicate transformers.

this paper, is the analysis of notions of approximation aiming at explaining and justifying (order-theoretic) properties of categories of domains. For example, in [Fio94c, Fio94a], while studying the interaction between partiality and order-enrichment we considered contextual approximation which, in the framework we were working in, coincided with the specialisation preorder . But in the applications carried out in [FP94, Fio94a] we had to work with an axiomatised notion of approximation, instead of the aforementioned one, for the following two reasons: first, the specialisation preorder is not appropriate in categories of domains and stable functions (see [Fio94c]) and, second, we do not know of non-order-theoretic axioms making the specialisation preorder !-complete. To overcome these drawbacks another notion of approximation was to be considered. And, it was the second problem that motivated the intensional notion of approximation provided by the path relation. In fact, it is shown in [Fio94b] that under suitable axioms the path relation can be equipped with a canonical passage-to-the-limit operator appropriate for fixed-point computations; stronger axioms make this operator be given by lubs of !-chains

By giving a translation from typed object calculus into Plotkin’s FPC, we demonstrate that every computationally sound and adequate model of FPC (with eager operational semantics), is also a sound and adequate model of typed object calculus. This establishes that denotational equality is contained in operational equivalence, i.e. that for any such model of typed object calculus, if two terms have equal denotations, then no program (or rather program context) can distinguish between those two terms. Hence we show that FPC models can be used in the study of program transformations (program algebra) for typed object calculus. Our treatment is based on self-application interpretation and subtyping is not considered. The typed object calculus under consideration is a variation of Abadi and Cardelli’s first-order calculus with sum and product types, recursive types, and the usual method update and method invocation in a form where the object types have assimilated the recursive types. As an additional result, we prove subject reduction for this calculus.