Advertisement

Adhesivity Is Not Enough: Local Church-Rosser Revisited

  • Paolo Baldan
  • Fabio Gadducci
  • Pawel Sobociński
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6907)

Abstract

Adhesive categories provide an abstract setting for the double-pushout approach to rewriting, generalising classical approaches to graph transformation. Fundamental results about parallelism and confluence, including the local Church-Rosser theorem, can be proven in adhesive categories, provided that one restricts to linear rules. We identify a class of categories, including most adhesive categories used in rewriting, where those same results can be proven in the presence of rules that are merely left-linear, i.e., rules which can merge different parts of a rewritten object. Such rules naturally emerge, e.g., when using graphical encodings for modelling the operational semantics of process calculi.

Keywords

Adhesive and extensive categories double-pushout rewriting local Church-Rosser property parallel and sequential independence 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baldan, P., Gadducci, F., Montanari, U.: Concurrent rewriting for graphs with equivalences. In: Baier, C., Hermanns, H. (eds.) CONCUR 2006. LNCS, vol. 4137, pp. 279–294. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  2. 2.
    Corradini, A., Montanari, U., Rossi, F., Ehrig, H., Heckel, R., Löwe, M.: Algebraic approaches to graph transformation I: Basic concepts and double pushout approach. In: Rozenberg, G. (ed.) Handbook of Graph Grammars and Computing by Graph Transformation, vol. 1, pp. 163–245. World Scientific, Singapore (1997)CrossRefGoogle Scholar
  3. 3.
    Corradini, A., Gadducci, F.: On term graphs as an adhesive category. In: TERMGRAPH 2004. ENTCS, vol. 127(5), pp. 43–56. Elsevier, Amsterdam (2005)Google Scholar
  4. 4.
    Drewes, F., Habel, A., Kreowski, H.-J.: Hyperedge replacement graph grammars. In: Rozenberg, G. (ed.) Handbook of Graph Grammars and Computing by Graph Transformation, vol. 1, pp. 95–162. World Scientific, Singapore (1997)CrossRefGoogle Scholar
  5. 5.
    Ehrig, H., Ehrig, K., Prange, U., Täntzer, G.: Fundamentals of Algebraic Graph Transformation. Monographs in Theoretical Computer Science. Springer, Heidelberg (2006)zbMATHGoogle 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., Habel, A., Parisi-Presicce, F.: Basic results for two types of high-level replacement systems. In: GETGRATS Closing Workshop. ENTCS, vol. 51, pp. 127–138. Elsevier, Amsterdam (2002)Google Scholar
  8. 8.
    Ehrig, H., König, B.: Deriving bisimulation congruences in the DPO approach to graph rewriting with borrowed contexts. Mathematical Structures in Computer Science 16(6), 1133–1163 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    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)Google Scholar
  10. 10.
    Gadducci, F.: Graph rewriting for the π-calculus. Mathematical Structures in Computer Science 17(3), 407–437 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Gadducci, F., Lluch Lafuente, A.: Graphical encoding of a spatial logic for the π-calculus. In: Mossakowski, T., Montanari, U., Haveraaen, M. (eds.) CALCO 2007. LNCS, vol. 4624, pp. 209–225. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  12. 12.
    Gadducci, F., Monreale, G.V.: A decentralized implementation of mobile ambients. In: Ehrig, H., Heckel, R., Rozenberg, G., Taentzer, G. (eds.) ICGT 2008. LNCS, vol. 5214, pp. 115–130. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  13. 13.
    Habel, A., Müller, J., Plump, D.: Double-pushout graph transformation revisited. Mathematical Structures in Computer Science 11(5), 637–688 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Heindel, T.: Hereditary pushouts reconsidered. In: Ehrig, H., Rensink, A., Rozenberg, G., Schürr, A. (eds.) ICGT 2010. LNCS, vol. 6372, pp. 250–265. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  15. 15.
    Johnstone, P.T., Lack, S., Sobociński, P.: Quasitoposes, quasiadhesive categories and artin glueing. In: Mossakowski, T., Montanari, U., Haveraaen, M. (eds.) CALCO 2007. LNCS, vol. 4624, pp. 312–326. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  16. 16.
    Lack, S., Sobociński, P.: Adhesive and quasiadhesive categories. Theoretical Informatics and Applications 39(3), 511–545 (2005)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2011

Authors and Affiliations

  • Paolo Baldan
    • 1
  • Fabio Gadducci
    • 2
  • Pawel Sobociński
    • 3
  1. 1.Dipartimento di Matematica Pura e ApplicataUniversità di PadovaItaly
  2. 2.Dipartimento di InformaticaUniversità di PisaItaly
  3. 3.School of Electronics and Computer ScienceUniversity of SouthamptonEngland

Personalised recommendations