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 reports early stages of this research and defines a semantic model, outlines how the model can be used as a semantic basis for current objectoriented design notations and gives a simple example of how the model facilitates program development. The paper is structured as follows: