Behavioural subtyping relations for object-oriented formalisms

Abstract. In this paper we investigate the object-oriented notion of subtyping in the context of behavioural formalisms. Subtyping in OO-formalisms is closely related to the concept of inheritance. The central issue in the choice of subtyping relations among classes is the principle of substitutability: an instance of the subtype should be usable wherever an instance of the supertype was expected. Depending on the interpre-tation of \usable", we obtain a variety of subtyping relations: stronger subtyping relations, allowing one to share the subtype instance among dierent clients without any change compared with the supertype, and weaker relations, restricting the possibilities of interference of dierent clients on the subtype instance. The subtyping relations are taxonom-ically ordered in a hierarchy. The concept of \usability " is formalised via testing scenarios, which provide alternative characterisations for the subtyping relations.