In typed object-oriented languages the subtype relation is typically based on the inheritance hierarchy. This approach, however, leads either to insecure type-systems or to restrictions on inheritance that make it less flexible than untyped Smalltalk inheritance. We present a new typed model of inheritance that allows more of the flexibility of Smalltalk inheritance within a statically-typed system. Significant features of our analysis are the introduction of polymorphism into the typing of inheritance and the uniform application of inheritance to objects, classes and types. The resulting notion of type inheritance allows us to show that the type of an inherited object is an inherited type but not always a subtype.

This paper develops a system of explicit interfaces for object-oriented programming. The system provides the benefits of module interfaces found in languages like Ada and Modula-2 while preserving the expressiveness that gives untyped object-oriented languages like Smalltalk their flexibility. Interfaces are interpreted as polymorphic types to make the system sufficiently powerful. We use interfaces to analyze the properties of inheritance, and identify three distinct kinds of inheritance in object-oriented programming, corresponding to objects, classes, and interfaces, respectively. Object interfaces clarify the distinction between interface containment and inheritance and give insight into limitations caused by equating the notions of type and class in many typed object-oriented programming languages. Interfaces also have practical consequences for design, specification, and maintenance of object-oriented systems.