Mappings, Maps and Tables: Towards Formal Semantics for Associations in UML2

  • Zinovy Diskin
  • Juergen Dingel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4199)


In fact, UML2 offers two related yet different definitions of associations. One is implicit in several Description and Semantics sections of the specification and belongs to the UML folklore. It simply says that an association is a set of links. The other – official and formal – definition is explicitly fixed by the UML metamodel and shows that there is much more to associations than just being sets of links. Particularly, association ends can be owned by either participating classes or by the very association (with a striking difference between binary and multiary associations), be navigable or not, and have some constraints on combining ownership and navigability.

The paper presents a formal framework, based on sets and mappings, where all notions involved in the both definitions can be accurately explained and formally explicated. Our formal definitions allow us to reconcile the two views of associations, unify ownership for binary and multiary associations and, finally, detect a few flaws in the association part of the UML2 metamodel.


Structural Mapping Formal Semantic Structural View Operational View Extension Table 


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  1. 1.
    Diskin, Z.: Visualization vs. specification in diagrammatic notations: A case study with the UML. In: Hegarty, M., Meyer, B., Narayanan, N.H. (eds.) Diagrams 2002. LNCS (LNAI), vol. 2317, pp. 112–115. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  2. 2.
    Diskin, Z., Kadish, B.: Variable set semantics for keyed generalized sketches: Formal semantics for object identity and abstract syntax for conceptual modeling. Data & Knowledge Engineering 47, 1–59 (2003)CrossRefGoogle Scholar
  3. 3.
    Génova, G., Llorens, J., Martínez, P.: Semantics of the minimum multiplicity in ternary associations in UML. In: Gogolla, M., Kobryn, C. (eds.) UML 2001. LNCS, vol. 2185, pp. 329–341. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  4. 4.
    Gunter, C.: Semantics of programming languages. MIT Press, Cambridge (1992)MATHGoogle Scholar
  5. 5.
    Hull, R., King, R.: Semantic database modeling: Survey, applications and research issues. ACM Computing Surveys 19(3), 201–260 (1987)CrossRefGoogle Scholar
  6. 6.
    Milicev, D.: On the semantics of associations and association ends in UML (submitted for publication)Google Scholar
  7. 7.
    Milicev, D., Selic, B., et al.: Joint E-mail Discussion, Fall (2005)Google Scholar
  8. 8.
    Object Management Group, Unified Modeling Language: Superstructure. version 2.0. Formal/05-07-04 (2005),
  9. 9.
    Rumbaugh, J., Jacobson, I., Booch, G.: The Unified Modeling Language Reference Manual, 2nd edn. Addison-Wesley, Reading (2004)Google Scholar
  10. 10.
    Selic, B.: Personal Communication, Fall (2005)Google Scholar
  11. 11.
    Stevens, P.: On the interpretation of binary associations in the unified modeling language. Software and Systems Modeling (1) (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Zinovy Diskin
    • 1
  • Juergen Dingel
    • 1
  1. 1.School of ComputingQueen’s UniversityKingstonCanada

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