This thesis develops a semantic model of inheritance and investigates its applications for the analysis and design of programming languages. Inheritance is a mechanism for incremental programming in the presence of self-reference. This interpretation of inheritance is formalized using traditional techniques of fixed-point theory, resulting in a compositional model of inheritance that is directly applicable to object-oriented languages. Novel applications of inheritance revealed by the model are illustrated to show that inheritance has wider significance beyond object-oriented class inheritance. Constraints induced by self-reference and inheritance are investigated using type theory and yield a formal characterization of abstract classes and a demonstration that the subtype relation is a direct consequence of the basic mechanism of inheritance. The model is proven equivalent to the operational semantics of inheritance embodied by the interpreters of object-oriented languages like Smalltalk. Concise descriptions of inheritance behavior in several object-oriented languages, including Smalltalk, Beta, Simula, and Flavors, are presented in a common framework that facilitates direct comparison of their features.

This paper presents a denotational model of inheritance. The model is based on an intuitive motivation of inheritance as a mechanism for deriving modified versions of recursive definitions. The correctness of the model is demonstrated by proving it equivalent to an operational semantics of inheritance based upon the method lookup algorithm of object-oriented languages.

We present and compare two models of object-oriented languages. The first we call the closure model because it uses closures to encapsulate side effects on objects, and accordingly makes the operations on an object a part of that object. It is shown that this denotational framework is adequate to explain classes, instantiation, and inheritance in the style of Simula as well as Smalltalk–80. The second we call the data structure model because it mimics the implementations of data structure languages like CLU in representing objects by records of instance variables, while keeping the operations on objects separate from the objects themselves. This yields a model which is very simple, at least superficially. Both the models are presented by way of a sequence of languages, culminating in a language with Smalltalk–80-style inheritance. The mathematical relationship between them is then discussed and it is shown that the models give equivalent results. It will emerge from this discussion that more appropriate names for the two models might be the fixed-point model and the self-application model. 1