A Calculus for Overloaded Functions with Subtyping ( extended abstract )

We present a simple extension of typed-calculus where functions can be overloaded by adding diier-ent \pieces of code". In short, the code of an overloaded function is formed by several branches of code; the branch to execute is chosen, when the function is applied, according to a particular selection rule which depends on the type of the argument. The crucial feature of the present approach is that a subtyping relation is deened among types, such that the type of a term generally decreases during computation, and this fact induces a distinction between the \compile-time" type and the \run-time" type of a term. We study the case of overloaded functions where the branch selection depends on the run-time type of the argument, so that overloading cannot be eliminated by a static analysis of code, but is an essential feature to be dealt with during computation. We obtain in this way a type-dependent calculus, which diiers from the various ?calculi where types do not play, essentially, any r^ ole during computation. We prove Connuence and Strong Normalization for this calculus as well as a generalized Subject-Reduction theorem The deenition of this calculus is driven by the understanding of object-oriented features and the connections between our calculus and object-orientedness are extensively stressed. We show that this calculus provides a foundation for typed object-oriented languages which solves some of the problems of the standard record-based approach. It also provides a type-discipline for a relevant fragment of the \core frame-work" (see Kee89]) of CLOS. Permission of copy all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish requires a fee and/or speciic permission 1 Introduction.