A Theory of Class

Abstract

We present a mathematical theory of class. The theory is general, in that it encompasses many different approaches to type abstraction, such as type constructors, generic parameters, classes, inheritance and polymorphism. The theory is elegant, in that it is based on a simple generalisation of F-bounds. The theory has timely implications for emerging OMG standards and future language designs.

The Theory of Classification, Part 13: Template Classes and Genericity.

“…So, we have successfully demonstrated the removal of genericity during the operation of inheritance. This close integration of generic classes with inheritance and with old-fashioned type constructors, like Pascal's SET OF... was first demonstrated by Simons [16,17], who also showed the important formal property of confluence. This property allows the same type to be derived either by instantiating, then inheriting; or by inheriting, then instantiating the parameters, and is an important symmetry property.…”

“…It is particularly satisfying to find that F-bounds can also be used to model constrained generic types [17,18]. To define the SortedList above, we first need to define a generator for the Comparable class, assuming that this supplies the methods lessThan and equal: GenComparable = λσ.…”

“…Cook originally used F-bounds just to model the self-types of classes and explain how these were modified under inheritance [15]. Simons integrated this use of F-bounds with generic classes in his Theory of Classification [16,17], finding that the same modelling concept could be used everywhere. In a later paper, he also showed how all three of Eiffel's typing mechanisms (conformance, type anchors and constrained genericity) could be modelled by F-bounds, demonstrating the economy and power of the theory [18].…”