Foundations for Virtual Types

. Virtual types have been proposed as a notation for generic programming in object-oriented languages---an alternative to the more familiar mechanism of parametric classes. The tradeoffs between the two mechanisms are a matter of current debate: for many examples, both appear to offer convenient (indeed almost interchangeable) solutions; in other situations, one or the other seems to be more satisfactory. However, it has proved difficult to draw rigorous comparisons between the two approaches, partly because current proposals for virtual types vary considerably in their details, and partly because the proposals themselves are described rather informally, usually in the complicating context of full-scale language designs. Work on the foundations of object-oriented languages has already established a clear connection between parametric classes and the polymorphic functions found in familiar typed lambda-calculi. Our aim here is to explore a similar connection between virtua..