. We present a method for providing semantic interpretations for languages with a type system featuring inheritance polymorphism. Our approach is illustrated on an extension of the language Fun of Cardelli and Wegner, which we interpret via a translation into an extended polymorphic lambda calculus. Our goal is to interpret inheritances in Fun via coercion functions which are definable in the target of the translation. Existing techniques in the theory of semantic domains can be then used to interpret the extended polymorphic lambda calculus, thus providing many models for the original language. This technique makes it possible to model a rich type discipline which includes parametric polymorphism and recursive types as well as inheritance. A central difficulty in providing interpretations for explicit type disciplines featuring inheritance in the sense discussed in this paper arises from the fact that programs can type-check in more than one way. Since interpretations follow the type...

Abstract not found

Functional programmers often reason about programs as if they were written in a total language, expecting the results to carry over to non-total (partial) languages. We justify such reasoning.

We investigate the development of theories of types and computability via realizability.

. We introduce a bicartesian closed category of what we call profinite domains. Study of these domains is carried out through the use of an equivalent category of pre-orders in a manner similar to the information systems approach advocated by Dana Scott and others. A class of universal profinite domains is defined and used to derive sufficient conditions for the profinite solution of domain equations involving continuous operators. As a special instance of this construction, a universal domain for the category SFP is demonstrated. Necessary conditions for the existence of solutions for domain equations over the profinites are also given and used to derive results about solutions of some equations. A new universal bounded complete domain is also demonstrated using an operator which has bounded complete domains as its fixed points. 1 Introduction. For our purposes a domain equation has the form X ¸ = F (X) where F is an operator on a class of semantic domains (typically, F is an endof...

This paper extends Curry-Howard interpretations of Intuitionistic Logic (IL) and Intuitionistic Linear Logic (ILL) with rules for recursion. The resulting term languages, the rec -calculus and the linear rec -calculus respectively, are given sound categorical interpretations. The embedding of proofs of IL into proofs of ILL given by the Girard Translation is extended with the rules for recursion, such that an embedding of terms of the rec -calculus into terms of the linear rec -calculus is induced via the extended Curry-Howard isomorphisms. This embedding is shown to be sound with respect to the categorical interpretations. Full version of paper to appear in Proceedings of CSL '94, LNCS 933, 1995. y Basic Research in Computer Science, Centre of the Danish National Research Foundation. Contents 1 Introduction 4 2 The Categorical Picture 6 2.1 Previous Work and Related Results : : : : : : : : : : : : : : : : : : : : : : 6 2.2 How to deal with parameters : : : : : : : ...

We investigate the duality between call-by-name recursion and call-by-value iteration on the -calculi. The duality between call-by-name and call-by-value was first studied by Filinski, and Selinger has studied the category-theoretic duality on the models of the call-by-name -calculus and the call-by-value one. We extend the call-by-name -calculus and the call-by-value one with a fixed-point operator and an iteration operator, respectively. We show that the dual translations constructed by Selinger can be expanded into our extended -calculi, and we also discuss their implications to practical applications.

We motivate and define a category of topological domains, whose objects are certain topological spaces, generalising the usual ω-continuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also supports the construction of free algebras for (in)equational theories, can be used as the basis for a theory of computability, and provides a model of parametric polymorphism.

We present a domain-theoretic model of parametric polymorphism based on admissible per’s over a domain-theoretic model of the untyped lambda calculus. The model is shown to be a model of Abadi & Plotkin’s logic for parametricity, by the construction of an LAPL-structure as defined by the authors in [7, 5]. This construction gives formal proof of solutions to a large class of recursive domain equations, which we explicate. As an example of a computation in the model, we explicitly describe the natural numbers object obtained using parametricity. The theory of admissible per’s can be considered a domain theory for (impredicative) polymorphism. By studying various categories of admissible and chain complete per’s and their relations, we discover a picture very similar to that of domain theory. 1

. A new games model of the language FPC, a type theory with products, sums, function spaces and recursive types, is described. A definability result is proved, showing that every finite element of the model is the interpretation of some term of the language. 1. Introduction. The work of Lorenzen [24, 23] proposed dialogue games as a foundation for intuitionistic logic. The idea is simple: associated to a formula A is a set of moves for two players, each of which is either an attack on A---an attempt to refute its validity---or a defence. The players, O who wants to refute A and P who wants to prove A, take turns to make moves according to some rules. The rules determine which player has won when play ends, and the formula A is semantically valid if there is a strategy by which P can always win: a winning strategy. More recently, games of this kind have been applied in computer science to give programming languages a new kind of semantics with a strong intensional flavour. The game in...