We construct a realizability model of recursive polymorphic types, starting from an untyped language of terms and contexts. An orthogonality relation e # indicates when a term e and a context # may be safely combined in the language. Types are interpreted as sets of terms closed by biorthogonality. Our main result states that recursive types are approximated by converging sequences of interval types. Our proof is based on a "type-directed" approximation technique, which departs from the "language-directed" approximation technique developed by MacQueen, Plotkin and Sethi in the ideal model. We thus keep the language elementary (a call-by-name #-calculus) and unstratified (no typecase, no reduction labels). We also include a short account of parametricity, based on an orthogonality relation between quadruples of terms and contexts.

We present a generalization of the ideal model for recursive polymorphic types. Types are defined as sets of terms instead of sets of elements of a semantic domain. Our proof of the existence of types (computed by fixpoint of a typing operator) does not rely on metric properties, but on the fact that the identity is the limit of a sequence of projection terms. This establishes a connection with the work of Pitts on relational properties of domains. This also suggests that ideals are better understood as closed sets of terms defined by orthogonality with respect to a set of contexts.

In a previous paper we have defined a semantic preorder called operational subsumption, which compares terms according to their error generation behaviour. Here we apply this abstract framework to a concrete language, namely the Abadi-Cardelli object calculus. Unlike most semantic studies of objects, which deal with typed equalities and therefore require explicitly typed languages, we start here from a untyped world. Type inference is introduced in a second step, together with an ideal model of types and subtyping. We show how this approach flexibly accommodates for several variants, and finally propose a novel semantic interpretation of structural subtyping as embedding-projection pairs. 1 Introduction In a previous paper [10] we have defined a semantic preorder called operational subsumption, which compares terms according to their error generation behaviour. Together with the technical device of labeled reductions, used as a syntactic characterization of finite approximations, thi...