Local Confluence for Rules with Nested Application Conditions

  • Hartmut Ehrig
  • Annegret Habel
  • Leen Lambers
  • Fernando Orejas
  • Ulrike Golas
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6372)

Abstract

Local confluence is an important property in many rewriting and transformation systems. The notion of critical pairs is central for being able to verify local confluence of rewriting systems in a static way. Critical pairs are defined already in the framework of graphs and adhesive rewriting systems. These systems may hold rules with or without negative application conditions. In this paper however, we consider rules with more general application conditions — also called nested application conditions — which in the graph case are equivalent to finite first-order graph conditions. The classical critical pair notion denotes conflicting transformations in a minimal context satisfying the application conditions. This is no longer true for combinations of positive and negative application conditions — an important special case of nested ones — where we have to allow that critical pairs do not satisfy all the application conditions. This leads to a new notion of critical pairs which allows to formulate and prove a Local Confluence Theorem for the general case of rules with nested application conditions. We demonstrate this new theory on the modeling of an elevator control by a typed graph transformation system with positive and negative application conditions.

Keywords

Critical pairs local confluence nested application conditions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)Google Scholar
  2. 2.
    Ehrig, H., Ehrig, K., Prange, U., Taentzer, G.: Fundamentals of Algebraic Graph Transformation. In: EATCS Monographs of Theoretical Computer Science. Springer, Heidelberg (2006)Google Scholar
  3. 3.
    Knuth, N.E., Bendix, P.B.: Simple word problems in universal algebra. In: Leech, J. (ed.) Computational Problems in Abstract Algebra, pp. 263–297 (1970)Google Scholar
  4. 4.
    Plump, D.: Hypergraph rewriting: Critical pairs and undecidability of confluence. In: Term Graph Rewriting: Theory and Practice, pp. 201–213. John Wiley, Chichester (1993)Google Scholar
  5. 5.
    Plump, D.: Confluence of graph transformation revisited. In: Middeldorp, A., van Oostrom, V., van Raamsdonk, F., de Vrijer, R. (eds.) Processes, Terms and Cycles: Steps on the Road to Infinity. LNCS, vol. 3838, pp. 280–308. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  6. 6.
    Habel, A., Heckel, R., Taentzer, G.: Graph grammars with negative application conditions. Fundamenta Informaticae 26, 287–313 (1996)MATHMathSciNetGoogle Scholar
  7. 7.
    Lambers, L., Ehrig, H., Prange, U., Orejas, F.: Embedding and confluence of graph transformations with negative application conditions. In: Ehrig, H., Heckel, R., Rozenberg, G., Taentzer, G. (eds.) ICGT 2008. LNCS, vol. 5214, pp. 162–177. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  8. 8.
    Habel, A., Pennemann, K.H.: Correctness of high-level transformation systems relative to nested conditions. Mathematical Structures in Computer Science 19, 245–296 (2009)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Courcelle, B.: The expression of graph properties and graph transformations in monadic second-order logic. In: Handbook of Graph Grammars and Computing by Graph Transformation, vol. 1, pp. 313–400. World Scientific, Singapore (1997)CrossRefGoogle Scholar
  10. 10.
    Lambers, L., Ehrig, H., Habel, A., Orejas, F., Golas, U.: Local Confluence for Rules with Nested Application Conditions based on a New Critical Pair Notion. Technical Report 2010-7, Technische Universität Berlin (2010)Google Scholar
  11. 11.
    Lack, S., Sobociński, P.: Adhesive categories. In: Walukiewicz, I. (ed.) FOSSACS 2004. LNCS, vol. 2987, pp. 273–288. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  12. 12.
    Newman, M.H.A.: On theories with a combinatorial definition of “equivalence”. Annals of Mathematics (43,2), 223–243 (1942)Google Scholar
  13. 13.
    Koch, M., Mancini, L.V., Parisi-Presicce, F.: Graph-based specification of access control policies. Journal of Computer and System Sciences 71, 1–33 (2005)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Ehrig, H., Ehrig, K., Habel, A., Pennemann, K.H.: Theory of constraints and application conditions: From graphs to high-level structures. Fundamenta Informaticae 74(1), 135–166 (2006)MATHMathSciNetGoogle Scholar
  15. 15.
    Ehrig, H., Habel, A., Lambers, L.: Parallelism and concurrency theorems for rules with nested application conditions. In: Essays Dedicated to Hans-Jörg Kreowski on the Occasion of His 60th Birthday. Electronic Communications of the EASST, vol. 26 (2010)Google Scholar
  16. 16.
    Orejas, F., Ehrig, H., Prange, U.: A logic of graph constraints. In: Fiadeiro, J.L., Inverardi, P. (eds.) FASE 2008. LNCS, vol. 4961, pp. 179–198. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  17. 17.
    Pennemann, K.H.: Resolution-like theorem proving for high-level conditions. In: Ehrig, H., Heckel, R., Rozenberg, G., Taentzer, G. (eds.) ICGT 2008. LNCS, vol. 5214, pp. 289–304. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  18. 18.
    Pennemann, K.H.: An algorithm for approximating the satisfiability problem of high-level conditions. In: Proc. Int. Workshop on Graph Transformation for Verification and Concurrency (GT-VC 2007). ENTCS, vol. 213, pp. 75–94 (2008), http://formale-sprachen.informatik.uni-oldenburg.de/pub/index.html
  19. 19.
    Lambers, L., Ehrig, H., Orejas, F.: Conflict detection for graph transformation with negative application conditions. In: Corradini, A., Ehrig, H., Montanari, U., Ribeiro, L., Rozenberg, G. (eds.) ICGT 2006. LNCS, vol. 4178, pp. 61–76. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  20. 20.
    Rensink, A.: Representing first-order logic by graphs. In: Ehrig, H., Engels, G., Parisi-Presicce, F., Rozenberg, G. (eds.) ICGT 2004. LNCS, vol. 3256, pp. 319–335. Springer, Heidelberg (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Hartmut Ehrig
    • 1
  • Annegret Habel
    • 2
  • Leen Lambers
    • 3
  • Fernando Orejas
    • 4
  • Ulrike Golas
    • 1
  1. 1.Technische Universität BerlinGermany
  2. 2.Carl v. Ossietzky Universität OldenburgGermany
  3. 3.Hasso Plattner InstitutUniversität PotsdamGermany
  4. 4.Universitat Politècnica de CatalunyaSpain

Personalised recommendations