Categorical Foundations of Distributed Graph Transformation

  • Hartmut Ehrig
  • Fernando Orejas
  • Ulrike Prange
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4178)


A distributed graph (N,D) consists of a network graph N and a commutative diagram D over the scheme N which associates local graphs D(n i ) and graph morphisms D(e): D(n 1) →D(n 2) to nodes n 1, n 2 and edges e: n 1n 2 in N.

Although there are several interesting applications of distributed graphs and transformations, even the basic pushout constructions for the double pushout approach of distributed graph transformation could be shown up to now only in very special cases.

In this paper we show that the category of distributed graphs can be considered as a Grothendieck category over a specific indexed category, which assigns to each network N the category of all diagrams D of shape N. In this framework it is possible to give a free construction which allows to construct for each diagram D 1 over N 1 and network morphism h:N 1N 2 a free extension F h (D 1) over N 2 and to show that the Grothendieck category is complete and cocomplete if the underlying category of local graphs has these properties.

Moreover, an explicit construction for general pushouts of distributed graphs is given. This pushout construction is based on the free construction. The non-trivial proofs for free constructions and pushouts are the main contributions of this paper and they are compared with the special cases known up to now.


Natural Transformation Graph Transformation Network Graph Local Graph Shared Part 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Taentzer, G.: Distributed Graphs and Graph Transformation. Applied Categorical Structures 7(4), 431–462 (1999)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bottoni, P., Parisi-Presicce, F., Taentzer, G.: Specifying Integrated Refactoring with Distributed Graph Transformations. In: Pfaltz, J.L., Nagl, M., Böhlen, B. (eds.) AGTIVE 2003. LNCS, vol. 3062, pp. 220–235. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  3. 3.
    Bottoni, P., Parisi-Presicce, F., Taentzer, G., Pulcini, S.: Maintaining Coherence between Models with Distributed Rules: From Theory to Eclipse. In: Bruni, R., Varró, D. (eds.) Proc. of GT-VMT 2006. ENTCS, pp. 81–91. Elsevier, Amsterdam (2006)Google Scholar
  4. 4.
    Fiadeiro, J.: Categories for Software Engineering. Springer, Heidelberg (2006)Google Scholar
  5. 5.
    Goguen, J.: Sheaf Semantics for Concurrent Interacting Objects. Mathematical Structures in Computer Science 2(2), 159–191 (1992)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Ehrig, H., Habel, A., Padberg, J., Prange, U.: Adhesive High-Level Replacement Categories and Systems. In: Ehrig, H., Engels, G., Parisi-Presicce, F., Rozenberg, G. (eds.) ICGT 2004. LNCS, vol. 3256, pp. 144–160. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  7. 7.
    Ehrig, H., Prange, U., Taentzer, G.: Fundamental Theory for Typed Attributed Graph Transformation. In: Ehrig, H., Engels, G., Parisi-Presicce, F., Rozenberg, G. (eds.) ICGT 2004. LNCS, vol. 3256, pp. 161–177. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  8. 8.
    Ehrig, H., Ehrig, K., Prange, U., Taentzer, G.: Fundamentals of Algebraic Graph Transformation. Springer, Heidelberg (2006)MATHGoogle Scholar
  9. 9.
    Mac Lane, S.: Categories for the Working Mathematician. Graduate Texts in Mathematics, vol. 5. Springer, New York (1971)Google Scholar
  10. 10.
    Ehrig, H., Orejas, F., Prange, U.: Categorical Foundations of Distributed Graph Transformation: Long Version. Technical report, TU Berlin (2006)Google Scholar
  11. 11.
    Tarlecki, A., Burstall, R., Goguen, J.: Some Fundamental Algebraic Tools for the Semantics of Computation: Part 3: Indexed Categories. Theoretical Computer Science 91(2), 239–264 (1991)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Goguen, J.: Information Integration in Institutions. In: Moss, L. (ed.) Jon Barwise Memorial Volume. Indiana University Press (to appear, 2006)Google Scholar
  13. 13.
    Ehrig, H., Baldamus, M., Orejas, F.: New Concepts for Amalgamation and Extension in the Framework of Specification Logics. Technical Report 91/05, TU Berlin (1991)Google Scholar
  14. 14.
    Ehrig, H., Baldamus, M., Cornelius, F., Orejas, F.: Theory of Algebraic Module Specification including Behavioural Semantics, Constraints an Aspects of Generalized Morphisms. In: Nivat, M., Rattray, C., Rus, T., Scollo, G. (eds.) Invited Lecture Proc. of AMAST 1991, pp. 145–172. Springer, Heidelberg (1991)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Hartmut Ehrig
    • 1
  • Fernando Orejas
    • 2
  • Ulrike Prange
    • 1
  1. 1.Technical University of BerlinGermany
  2. 2.Technical University of CataloniaSpain

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