Patterns and Rules for Behavioural Subtyping

Subtyping relations for object-oriented formalisms describe relationships between superand subclasses which satisfy the substitutability requirement imposed on types and their subtypes. Behavioural subtyping is concerned with subtypes for active classes with an explicit dynamic behaviour, specifiable for instance by object-oriented formal methods combining state-based with behavioural formalisms. In this paper we develop syntactic patterns and semantic rules for the statebased part of a subclass which guarantee that the subclass is a behavioural subtype of its superclass. This allows to check for subtypes without computing the behavioural semantics of the class at all. Our results are similar to the ones linking data refinement in state-based methods with failure-divergence refinement in CSP. In contrast to classical data refinement, subtyping has to cope with additional new operations in the subclass.