A Calculus for Overloaded Functions with Subtyping

We present a simple extension of typed ?-calculus where functions can be overloaded by putting different "branches of code" together. When the function is applied, the branch to execute is chosen according to a particular selection rule which depends on the type of the argument. The crucial feature of the present approach is that the branch selection depends on the "run-time type" of the argument, which may differ from its compile-time type, because of the existence of a subtyping relation among types. Hence overloading cannot be eliminated by a static analysis of code, but it is an essential feature to be dealt with during computation. We obtain in this way a type-dependent calculus, which differs from the various ?-calculi where types to not play any role during computation. We prove confluence and a generalized subject-reduction theorem for this calculus. We prove strong normalization for a "stratified" subcalculus. The definition of this calculus is guided by the understanding of object-oriented features and the connections between our calculus and object-orientedness are extensive stressed. We show that this calculus provides a foundation for types object-oriented languages which solves some of the problems of the standard record-based approach.