Mathematics of UML

Abstract

It is commonly recognized that there are a few serious drawbacks in UML. Among the most often cited are (i) excessive size and complexity, (ii) limited customizability, (iii) unclear semantics. Less common is the understanding that these drawbacks are almost inevitable for UML due to the fundamental gaps in its logical foundations. Finally, it is barely known that the first of them instantly disappears, the second one becomes readily manageable and the third approachable, as soon as UML is based on a proper mathematical foundation already developed in mathematical category theory. One more, and very important, benefit is that categorical treatment of UML-models leads to a natural and simple notion of model mapping, and hence provides all the necessary prerequisites for efficient model management (cf. [BHP00]). In contrast, (iv) UML’s model mappings are hardly definable in a manageable way due to (i) (and (iii) too), and it is a really serious problem still not recognized in the UML literature. The goal of the present paper is to manifest these claims, outline general mathematical mechanisms that do the job, and demonstrate how they work with simple examples.

The craft of building information systems needs a shared language.... A language of these qualities may be a long way off, though UML2 will certainly make a step closer. The authors here debate the size and direction of this next step; another step will surely follow.

Joaquin Miller, “What UML should be”,the Guest Editor’s foreword to UML Panel in Comm.ACM [M2002]

The work was partially funded by the Latvian Council of Science.