A formal definition for the semantics of the whole-part (WP) relationship in the unified modeling language (UML) is proposed in this paper.

The notion of WP relationships in UML is defined in terms of the semantics of aggregation and composition. To make their case, the authors demonstrate the ambiguity attributed to the semantics of aggregation and composition, which has resulted in UML being incomplete, inconsistent, and incorrect. To resolve this ambiguity problem, the authors first identify a set of properties (primary or essential, and secondary or differential), which is manifested by WP relationships. An example of an essential property is asymmetry. An example of a differential property is transitivity. Next, the authors formalize a subset of these properties with a set of logical assertions.

The main contributions of this paper are the identification of meta-elements (or features), which has been used as a base to change the key relationships; and the formal specification of the meta-elements, which has resulted in providing a formal, mathematical semantics for diagrammatic description techniques, namely aggregation and composition.

The work provides a framework that reflects a change that is significant to the core theory of WP relationships. To this end, the framework provides not only a precise interpretation of WP relationships, but also sufficient explanations to justify its adoption in UML 2.0. The authors’ work is detailed and enjoyable. I would recommend this paper to anyone who is interested in the formalization of UML.