Lambda the Ultimate

Reading Martin Oderski's paper Scalable Component Abstractions, I ran across citations to the languages CeasarJ and Gbeta, both realizing a concept called virtual classes. This concept is explained, and a calculus given in the paper A Virtual Class Calculus by Erik Ernst, Klaus Ostermann, and William R. Cook.

Virtual classes are class-valued attributes of objects. Like virtual methods, virtual classes are defined in an object's class and may be redefined within subclasses. They resemble inner classes, which are also defined within a class, but virtual classes are accessed through object instances, not as static components of a class. When used as types, virtual classes depend upon object identity -- each object instance introduces a new family of virtual class types. Virtual classes support large-scale program composition techniques, including higher-order hierarchies and family polymorphism. The original definition of virtual classes in BETA left open the question of static type safety, since some type errors were not caught until runtime. Later the languages Caesar and gbeta have used a more strict static analysis in order to ensure static type safety. However, the existence of a sound, statically typed model for virtual classes has been a long-standing open question. This paper presents a virtual class calculus, VC, that captures the essence of virtual classes in these full-fledged programming languages. The key contributions of the paper are a formalization of the dynamic and static semantics of VC and a proof of the soundness of VC.