Behavioural Subtyping for a Type-theoretic Model of Objects

We present a reenement of the existential object model of Pierce and Turner PT94]. In addition to signatures (or interfaces) as the types of objects, we also provide classes as the types of objects. These class types not only specify an interface, but also a particular implementation. We show that class types can be interpreted in the standard PER model. Our main result is that the standard interpretation of subtyping in PER models { i.e. subtypes are subpers { is then behavioural subtyping in the sense of Leavens Lea90]. 1 Introduction Like most descriptions of objects in typed lambda calculus or typed object calculus, the existential object model of Pierce and Turner PT94] provides signatures or interfaces as the types of objects, and provides the usual syntactic notion of subtyping on these types. We consider a more reened notion of class type as the type of an object. A class type is a subtype of an interface type, speciied by a particular implementation and initial state. The existential object encoding can be carried out in F ! , but the class types we want to consider are not encodable in F !. Therefore we will deene a simple object-oriented functional language OO , that is essentially just some syntactic sugar for the existential object encoding, but extended with a notion of class type. The standard model of F ! is a PER model, which interprets types as partial equivalence relations (pers), and interprets subtyping as subset inclusion between pers BL90]. We show that class types can be interpreted in the PER model. For these interpretations of class types the subset relation turns out to be equivalent to the notion of behavioural sub-typing as deened by Leavens Lea90]. It is important to note that class types can be behavioural subtypes even though they give completely unrelated implementations. Although class types correspond to particular implementations, behavioural subtyping between class types