. The idea of parametric polymorphism is that of a single operator that can be used for different data types and whose behaviour is somehow uniform for each type. Reynolds [Reynolds, 1983] uses binary relations to define a uniformity condition for parametric polymorphism. In [Plotkin & Abadi, 1993] the authors proposed a second order logic for second order lambda-calculus; this logic is able to handle parametric polymorphism in the binary relational sense of Reynolds. In this paper we examine a categorical framework for this logic. This framework is based on the notion of categorical model of second order lambda-calculus as given, for example, in [Pitts, 1987, Seely, 1987, Robinson, 1992, Jacobs, 1991]. Going through the categorical constructions of the model, an unexpected property of quantification over type variables appears. A simple categorical calculation indicates what is the appropriate way to obtain the right adjoint to weakening that models universal quantification. The resul...

We present a denotational model for F < , the extension of second-order lambda calculus with subtyping defined in [Cardelli Wegner 1985]. Types are interpreted as arbitrary cpos and elements of types as natural transformations. We prove the soundness of our model with respect to the equational theory of F < [Cardelli et al. 1991] and show coherence. Our model is of independent interest, because it integrates ad-hoc and parametric polymorphism in an elegant fashion, admits nontrivial records and record update operations, and formalizes an "order faithfulness" criterion for well-behaved multiple subtyping. 0 Introduction A variety of denotational models for F < (the second--order lambda calculus with subtyping and bounded quantification) [Cardelli Wegner 1985] have already been proposed [MacQueen et al. 1986, Bruce et al. 1990, Bruce Longo 1990, Amadio 1991, Cardone 1989, Abadi Plotkin 1990, Bruce Mitchell 1992, Breazu--Tannen et al. 1991], and this paper presents yet one more. Our rea...

this document that certain implications are not based on a well stated formal theory but require a certain amount of hand-waving.

There are a number of applied lambda-calculi in which terms and types are annotated with parameters denoting either locations or locations in machine memory. Such calculi have been designed with safe memory-management operations in mind. It is difficult to construct directly denotational models for existing calculi of this kind. We approach the problem differently, by starting from a class of mathematical models that describe some of the essential semantic properties intended in these calculi. In particular, disjointness conditions between regions (or locations) are implicit in many of the memory-management operations. Bunched polymorphism provides natural type-theoretic mechanisms for capturing the disjointness conditions in such models. We illustrate this by adding regions to the basic disjointness model of αλ, the lambda-calculus associated to the logic of bunched implications. We show how both additive and multiplicative polymorphic quantifiers arise naturally in our models. A locations model is a special case. In order to relate this enterprise back to previous work on memory-management, we provide an example in which the model is refined and used to provide a denotational semantics for a language with explicit allocation and disposal of regions.

There are a number of applied lambda-calculi in which terms and types are annotated with parameters denoting either locations or locations in machine memory. Such calculi have been designed with safe memory-management operations in mind. It is difficult to construct directly denotational models for existing calculi of this kind. We approach the problem differently, by starting from a class of mathematical models that describe some of the essential semantic properties intended in these calculi. In particular, disjointness conditions between regions (or locations) are implicit in many of the memory-management operations. Bunched polymorphism provides natural type-theoretic mechanisms for capturing the disjointness conditions in such models. We illustrate this by adding regions to the basic disjointness model of αλ, the lambda-calculus associated to the logic of bunched implications. We show how both additive and multiplicative polymorphic quantifiers arise naturally in our models. A locations model is a special case. In order to relate this enterprise back to previous work on memory-management, we provide an example in which the model is refined and used to provide a denotational semantics for a language with explicit allocation and disposal of regions.