A categorical model of object-oriented systems is proposed and denoted using a -calculus. The model is used to provide a definition of design refinement. An example system is rigorously developed in Java from an initial user requirements by refining an initial design. Contents 1 Introduction 1 2 Object-Oriented Design Features 2 3 Object Behaviour 3 3.1 Object States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.2 Object Calculation Graphs . . . . . . . . . . . . . . . . . . . . . 4 3.3 Object-Oriented Designs . . . . . . . . . . . . . . . . . . . . . . . 6 3.4 Object Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.5 Behaviour Refinement . . . . . . . . . . . . . . . . . . . . . . . . 7 4 Object-Oriented Design Notation 8 4.1 Behaviour Functions . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.2 Example Behaviour Functions . . . . . . . . . . . . . . . . . . . . 10 4.3 Message Handling . . . . . . . . . . . . . . . . . . . . . . . . . . 11 ...

Current graphical object-oriented design notations are syntax-bound and semantic-free since they tend to focus on design representation rather than on the meaning of the design. This paper proposes a meaning for object-oriented designs in terms of object behaviours represented as constructions in category theory. A new design language is proposed, based on -notation, whose semantics is given by object behaviours. An example application is constructed as both a graphical design and using the design language. Submitted to BCS FACS Journal January 1999. 1 Introduction Current object-oriented design notations such as OMT [Run91], Booch [Boo94] and UML [UML98] are syntax-bound and semantic-free in the sense that they typically employ a large and rigorously enforceable collection of construction rules, but rarely provide a model to explain what is being constructed. Whilst this omission clearly does not prevent such notations being used effectively in the development of object-orie...

This paper proposes a new calculus for expressing the behaviour of object-oriented systems. The semantics of the calculus is given in terms of operators from computational category theory. The calculus aims to span the gulf between abstract specification and concrete implementation of object-oriented systems using mathematically verifiable properties and transformations. The calculus is compositional and can be used to express the behaviour of partial system views. The calculus is used to specify, analyse and refine a simple case study. 1