The Theory of Classification, Part 16: Rules of Extension and the Typing of Inheritance

This is the sixteenth article in a regular series on object-oriented type theory for nonspecialists. Earlier articles have built up λ-calculus models of objects [1], classes [2], inheritance [3, 4] and generic template types [5]. Classification describes the way in which typed objects fit into a hierarchy of classes, which nest inside each other [3]. Inheritance is the short-hand mechanism for definining a subclass by extension from another class, specifying only the additions and modifications to the base class [4]. Previously, we have modelled the inheritance of type [3] and implementation [4] and combined both of these in a model of typed inheritance [6]. We showed how short-hand inheritance expressions can be simplified to yield a canonical subclass definition that is type compatible with the base superclass from which it was derived. Further aspects of inheritance have included method combination [7], mixin inheritance [8] and inheritance among generic classes [5].