Multiple Inheritance and Type System Design

It is now widely accepted that inheritance is not subtyping [5], and that ‘‘inheritance is a relationship between implementations, while conformity is a relationship between interfaces’’ [1]. But like most maxims, these two are something of an oversimplification. Inheritance: Mechanism or Relation? As realized in Simula and Smalltalk, inheritance is a mechanism for generating new pieces of program from old pieces of program. As such it is of tremendous value to those who write programs (as opposed to those who write papers). In contrast, subtyping is a relation between objects; whether or not one object’s type conforms to another object’s type depends on the interface of those objects but not on the code that created them. In this sense it is obvious that inheritance is not subtyping. But in a wider sense, inheritance gives rise to a relation that is similar to subtyping. Following Bruce[2] we will call this relation inh; figure 1 shows how inh is related to inheritance. Suppose that we have a piece of program A from which we construct through the mechanism of inheritance another piece of program AB. As Cook [4] has shown, A and AB must be thought of not as classes, but as class generators, i.e., as functions from classes to classes. In programming language terms, this means that A and AB