Relaxing Behavioural Inheritance

Abstract

Object-oriented (OO) inheritance allows the definition of families of classes in a hierarchical way. In behavioural inheritance, a strong version, it should be possible to substitute an object of a subclass for an object of its superclass without any observable effect on the system. Behavioural inheritance is related to formal refinement, but, as observed in the literature, the refinement constraints are too restrictive, ruling out many useful OO subclassings. This paper studies behavioural inheritance in the context of ZOO, an object-oriented style for Z. To overcome refinement's restrictions, this paper proposes relaxations to the behavioural inheritance refinement rules. The work is presented for Z, but the results are applicable to any OO language that supports design-by-contract.

Sound and Relaxed Behavioural Inheritance

“…This paper extends the work presented in [3,2], elaborating and reinforcing the paper's BI relaxations. This extension has four key components:…”

“…This paper investigates behavioural inheritance (BI) [28] by building up on insight gained from previous work [2,3]. It delves into BI's foundations through an abstract mathematical model of object orientation to come up with sound ways of reconciling the correctness-sensitive refinement facet of BI with the flexibility that characterises inheritance and the object-oriented (OO) paradigm.…”

“…This section presents the paper's OO mathematical model, drawn from ZOO [10,2,3], our approach to couch OO models in Z. The model rests on abstract data types (ADTs), enabling a connection to data refinement.…”

“…• The abstract model of OO of section 1.2 results from the insight gained from ZOO, a Z style of OO [2,10,3] that builds-up on Hall's work [19,20]. It represents a class as two ADTs, making it consistent with models of OO programming languages with a formal semantics (like those based on design-by-contract, such as Eiffel, JML and Spec ); Meyer [32] sees the OO paradigm founded on ADTs with classes having a type view and a module view, which correspond to the inner and outer ADTs that makeup a class in the paper's OO model.…”

“…The investigation is in the context of Z [39,24], a formal modelling language with a mature refinement theory [39,16]. The work builds up on ZOO, the OO style for Z presented in [10,2,3], that is the semantic domain of UML + Z [11,2,12] and the Visual Contract Language (VCL) for graphical modelling of software designs [7,8,9,6].…”