UML Class Diagram Classification Using Category Theory

Abstract

To effectively evaluate a system that performs operations on UML class diagrams, it is essential to cover a large variety of different types of diagrams. The coverage of the diagram space can be attained by grouping them into a finite number of disjoint equivalence classes, each containing diagrams that are structurally equivalent, and analyzing at least one diagram from each class. Currently, no formal method exists to implement this approach. In this paper, we describe an approach that implements this idea by mapping class diagrams into an appropriate UML category using category theory and defining isomorphism classes containing diagrams that are isomorphic. The main idea of our approach to partitioning the space of class diagrams is to consider that any UML class diagram is an instance of the UML metamodel, then identify a small number of basic template diagrams, carefully selected simple class diagrams with parameters/variables, and combine them into complex template diagrams using an operation based on the colimit from category theory. We demonstrate through experiments that almost all of a large set of class diagrams randomly generated from the UML metamodel can be classified as instances of the isomorphism classes generated through this combination operation.