Advertisement

On the Definition of Parallel Independence in the Algebraic Approaches to Graph Transformation

  • Andrea Corradini
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9946)

Abstract

Parallel independence between transformation steps is a basic and well-understood notion of the algebraic approaches to graph transformation, and typically guarantees that the two steps can be applied in any order obtaining the same resulting graph, up to isomorphism. The concept has been redefined for several algebraic approaches as variations of a classical “algebraic” condition, requiring that each matching morphism factorizes through the context graphs of the other transformation step. However, looking at some classical papers on the double-pushout approach, one finds that the original definition of parallel independence was formulated in set-theoretical terms, requiring that the intersection of the images of the two left-hand sides in the host graph is contained in the intersection of the two interface graphs. The relationship between this definition and the standard algebraic one is discussed in this position paper, both in the case of left-linear and non-left-linear rules.

Notes

Acknowledgments

The idea of spelling out the relationship between the standard algebraic and the pullback-based definitions of parallel independence maturated during stimulating discussions with Dominque Duval, Frédéric Prost, Rachid Echahed and Leila Ribeiro, during the work on the AGREE approach to GT. Hans-Jörg Kreowski provided me some references to the early literature on parallel independence. During the workshop where this work was presented, Michael Löwe suggested several technical improvements, including a new version of the last part of the proof of Proposition 2 that does not need partial maps classifiers: this will be presented in a forthcoming report.

References

  1. 1.
    Cockett, J., Lack, S.: Restriction categories II: partial map classification. Theor. Comput. Sci. 294(1–2), 61–102 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Corradini, A., Duval, D., Echahed, R., Prost, F., Ribeiro, L.: AGREE – algebraic graph rewriting with controlled embedding. In: Parisi-Presicce, F., Westfechtel, B. (eds.) ICGT 2015. LNCS, vol. 9151, pp. 35–51. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-21145-9_3 CrossRefGoogle Scholar
  3. 3.
    Corradini, A., Duval, D., Prost, F., Ribeiro, L.: Parallelism in AGREE transformations. In: Echahed, R., Minas, M. (eds.) ICGT 2016. LNCS, vol. 9761, pp. 37–53. Springer, Heidelberg (2016). doi: 10.1007/978-3-319-40530-8_3 CrossRefGoogle Scholar
  4. 4.
    Corradini, A., Ehrig, H., Löwe, M., Montanari, U., Rossi, F.: Abstract graph derivations in the double pushout approach. In: Schneider, H.J., Ehrig, H. (eds.) Graph Transformations in Computer Science. LNCS, vol. 776, pp. 86–103. Springer, Heidelberg (1994). doi: 10.1007/3-540-57787-4_6 CrossRefGoogle Scholar
  5. 5.
    Corradini, A., Heindel, T., Hermann, F., König, B.: Sesqui-Pushout rewriting. In: Corradini, A., Ehrig, H., Montanari, U., Ribeiro, L., Rozenberg, G. (eds.) ICGT 2006. LNCS, vol. 4178, pp. 30–45. Springer, Heidelberg (2006). doi: 10.1007/11841883_4 CrossRefGoogle Scholar
  6. 6.
    Corradini, A., Montanari, U., Rossi, F., Ehrig, H., Heckel, R., Löwe, M.: Algebraic approaches to graph transformation - part I: basic concepts and double pushout approach. In: Handbook of Graph Grammars and Computing by Graph Transformations, Foundations, vol. 1, pp. 163–246 (1997)Google Scholar
  7. 7.
    Danos, V., Heindel, T., Honorato-Zimmer, R., Stucki, S.: Reversible Sesqui-Pushout rewriting. In: Giese, H., König, B. (eds.) ICGT 2014. LNCS, vol. 8571, pp. 161–176. Springer, Heidelberg (2014). doi: 10.1007/978-3-319-09108-2_11 Google Scholar
  8. 8.
    Ehrig, H., Rosen, B.: Commutativity of Independent Transformations on Complex Objects. IBM Thomas J. Watson Research Division (1976)Google Scholar
  9. 9.
    Ehrig, H.: Introduction to the algebraic theory of graph grammars (a survey). In: Claus, V., Ehrig, H., Rozenberg, G. (eds.) Graph Grammars 1978. LNCS, vol. 73, pp. 1–69. Springer, Heidelberg (1979). doi: 10.1007/BFb0025714 CrossRefGoogle Scholar
  10. 10.
    Ehrig, H., Ehrig, K., Prange, U., Taentzer, G.: Fundamentals of Algebraic Graph Transformation. Monographs in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg (2006)zbMATHGoogle Scholar
  11. 11.
    Ehrig, H., Habel, A., Kreowski, H., Parisi-Presicce, F.: Parallelism and concurrency in high-level replacement systems. Math. Struct. Comput. Sci. 1(3), 361–404 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ehrig, H., Kreowski, H.-J.: Parallelism of manipulations in multidimensional information structures. In: Mazurkiewicz, A. (ed.) MFCS 1976. LNCS, vol. 45, pp. 284–293. Springer, Heidelberg (1976). doi: 10.1007/3-540-07854-1_188 CrossRefGoogle Scholar
  13. 13.
    Ehrig, H., Pfender, M., Schneider, H.J.: Graph-grammars: an algebraic approach. In: 14th Annual Symposium on Switching and Automata Theory, Iowa City, Iowa, USA, 15–17 October 1973, pp. 167–180. IEEE Computer Society (1973)Google Scholar
  14. 14.
    Kreowski, H.: Manipulation von Graphmanipulationen. Ph.D. thesis, Technische Universität, Berlin (1977)Google Scholar
  15. 15.
    Lack, S., Sobocinski, P.: Adhesive and quasiadhesive categories. Theor. Inf. Appl. 39(3), 511–545 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Löwe, M.: Graph rewriting in span-categories. In: Ehrig, H., Rensink, A., Rozenberg, G., Schürr, A. (eds.) ICGT 2010. LNCS, vol. 6372, pp. 218–233. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-15928-2_15 CrossRefGoogle Scholar
  17. 17.
    Rosen, B.K.: A Church-Rosser theorem for graph grammars. ACM SIGACT News 7(3), 26–31 (1975)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.Dipartimento di InformaticaUniversità di PisaPisaItaly

Personalised recommendations