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Double-Pullback Transitions and Coalgebraic Loose Semantics for Graph Transformation Systems

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Abstract

The classical algebraic approach to graph transformation is a mathematical theory based on categorical techniques with several interesting applications in computer science. In this paper, a new semantics of graph transformation systems (in the algebraic, double-pushout (DPO) approach) is proposed in order to make them suitable for the specification of concurrent and reactive systems. Classically, a graph transformation system comes with a fixed behavioral interpretation. Firstly, all transformation steps are intended to be completely specified by the rules of the system, that is, there is an implicit frame condition: it is assumed that there is a complete control about the evolution of the system. Hence, the interaction between the system and its (possibly unknown) environment, which is essential in a reactive system, cannot be modeled explicitly. Secondly, each sequence of transformation steps represents a legal computation of the system, and this makes it difficult to model systems with control. The first issue is addressed by providing graph transformation rules with a loose semantics, allowing for unspecified effects which are interpreted as activities of the environment. This is formalized by the notion of double-pullback transitions, which replace (and generalize) the well-known double-pushout diagrams by allowing for spontaneous changes in the context of a rule application. Two characterizations of double-pullback transitions are provided: the first one describes them in terms of extended direct DPO derivations, and the second one as incomplete views of parallel or amalgamated derivations. The issue of constraining the behavior of a system to transformation sequences satisfying certain properties is addressed instead by introducing a general notion of logic of behavioral constraints, which includes instances like start graphs, application and consistency conditions, and temporal logic constraints. The loose semantics of a system with restricted behavior is defined as a category of coalgebras over a suitable functor. Such category has a final object which includes all finite and infinite transition sequences satisfying the constraints.

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References

  1. Pfaltz, J. L. and Rosenfeld, A.: Web grammars, in Int. Joint Conference on Artificial Intelligence, 1969, pp. 609-619.

  2. Montanari, U.: Separable graphs, planar graphs and web grammars, Inform. and Control 16 (1970), 243-267.

    Google Scholar 

  3. Ehrig, H., Pfender, M. and Schneider, H.: Graph grammars: An algebraic approach, in 14th Annual IEEE Symposium on Switching and Automata Theory, IEEE, 1973, pp.167-180.

  4. Rozenberg, G. (ed.): Handbook of Graph Grammars and Computing by Graph Transformation, Volume 1: Foundations, World Scientific, 1997.

  5. Nagl, M. (ed.): Building Tightly Integrated Software Development Environments: The IPSEN Approach, LNCS 1170, Springer-Verlag, 1996.

  6. Schürr, A., Winter, A. and Zündorf, A.: Graph grammar engineering with PROGRES, in 5th European Software Engineering Conference (ESEC'95), Sitges, LNCS 989, Springer-Verlag, 1995, pp. 219-234.

  7. Engels, G., Heckel, R., Taentzer, G. and Ehrig, H.: A combined reference model-and viewbased approach to system specification, Internat. J. of Software and Knowledge Engrg. 7(4) (1997), 457-477.

    Google Scholar 

  8. Schürr, A.: Programmed graph replacement systems, in G. Rozenberg [4], pp. 479-546.

  9. Ermel, C., Rudolf, M. and Taentzer, G.: The AGG approach: Language and environment, in H. Ehrig et al. [10]. To appear.

  10. Ehrig, H., Engels, G., Kreowski, H.-J. and Rozenberg, G. (eds.): Handbook of Graph Grammars and Computing by Graph Transformation, Volume 2: Applications, Languages, and Tools, World Scientific, 1999. To appear.

  11. Ehrig, H.: Introduction to the algebraic theory of graph grammars, in V. Claus, H. Ehrig, and G. Rozenberg (eds.), 1st Graph Grammar Workshop, LNCS 73, Springer-Verlag, 1979, pp. 1-69.

  12. Corradini, A., Montanari, U., Rossi, F., Ehrig, H., Heckel, R. and Löwe, M.: Algebraic approaches to graph transformation, Part I: Basic concepts and double pushout approach, in G. Rozenberg [4], pp. 163-245.

  13. Ehrig, H. and Löwe, M.: Categorical principles, techniques and results for high-level replacement systems in computer science, Appl. Categorical Structures 1(1) (1993), 21-50.

    Google Scholar 

  14. Corradini, A., Ehrig, H., Löwe, M., Montanari, U. and Padberg, J.: The category of typed graph grammars and their adjunction with categories of derivations, in 5th Int. Workshop on Graph Grammars and their Application to Computer Science, Williamsburg '94, LNCS 1073, Springer-Verlag, 1996, pp. 56-74.

  15. Heckel, R., Corradini, A., Ehrig, H. and Löwe, M.: Horizontal and vertical structuring of typed graph transformation systems, Math. Structures Comput. Sci. 6(6) (1996), 613-648. Also Technical Report 96-22, TU Berlin.

    Google Scholar 

  16. Heckel, R.: Open graph transformation systems: A new approach to the compositional modelling of concurrent and reactive systems, PhD thesis, TU Berlin, 1998.

  17. Hoare, C.: An axiomatic basis for computer programming, Comm. ACM 12(10) (1986), 576-580.

    Google Scholar 

  18. Jones, C. B.: Systematic Software Development Using VDM, Prentice-Hall International, London, 1986.

    Google Scholar 

  19. Kreowski, H.-J. and Kuske, S.: On the interleaving semantics of transformation units-a step into GRACE, in 5th Int. Workshop on Graph Grammars and their Application to Computer Science, Williamsburg '94, LNCS 1073, Springer-Verlag, 1996, pp. 89-106.

  20. Spivey, J.: Understanding Z: A Specification Language and its Formal Semantics, Cambridge University Press, Cambridge, 1988.

    Google Scholar 

  21. Manna, Z. and Pnueli, A.: The Temporal Logic of Reactive and Concurrent Systems, Specification, Springer-Verlag, 1992.

  22. Harel, D.: Statecharts: A visual formalism for complex systems, Sci. Comput. Programming 8 (1987), 231-274.

    Google Scholar 

  23. Milner, R.: A Calculus for Communicationg Systems, LNCS 92, Springer-Verlag, 1980.

  24. Engels, G., Heckel, R., Taentzer, G. and Ehrig, H.: A view-oriented approach to system modelling using graph transformation, in Proc. of ESEC/FSE'97, Zürich, LNCS 1301, Springer-Verlag, 1997, pp. 327-343.

  25. Heckel, R.: Compositional verification of reactive systems specified by graph transformation, in Fundamental Approaches to Software Engineering, LNCS 1382, Springer-Verlag, 1998, pp. 138-153.

  26. Bauderon, M.: A category-theoretical approach to vertex replacement: The generation of in-finite graphs, in 5th Int. Workshop on Graph Grammars and their Application to Computer Science, Williamsburg '94, LNCS 1073, Springer-Verlag, 1996, pp. 27-37.

  27. Bauderon, M. and Jacquet, H.: Categorical product as a generic graph rewriting mechanism, Appl. Categorical Structures (1999). To appear. Also Technical Report 1166-97, University of Bordeaux.

  28. Engelfriet, J. and Rozenberg, G.: Node replacement graph grammars, in Rozenberg [4], pp. 1-94.

  29. Corradini, A., Montanari, U. and Rossi, F.: Graph processes, Fund. Inform. 26(3-4) (1996), 241-266.

    Google Scholar 

  30. Andries, M., Engels, G., Habel, A., Hoffmann, B., Kreowski, H.-J., Kuske, S., Plump, D., Schürr, A. and Taentzer, G.: Graph transformation for specification and programming, Sci. Comput. Programming (1999). To appear. Also Technical Report 7/96 of University of Bremen.

  31. Heckel, R., Engels, G., Ehrig, H. and Taentzer, G.: Classification and comparison of modularity concepts for graph transformation systems, in H. Ehrig et al. [10]. To appear.

  32. Ribeiro, L.: A telephone system's specification using graph grammars, Technical Report 96-23, Technical University of Berlin, 1996.

  33. Ehrig, H., Heckel, R., Llabres, M. and Orejas, F.: Construction and characterisation of doublepullback graph transitions, in Prelim. Proc. 6th Int. Workshop on Theory and Application of Graph Transformation (TAGT'98), Paderborn, 1998.

  34. Böhm, P., Fonio, H.-R. and Habel, A.: Amalgamation of graph transformations: a synchronization mechanism, J. Comput. System Sci. 34 (1987), 377-408.

    Google Scholar 

  35. Heckel, R.: Embedding of conditional graph transformations, in G. Valiente Feruglio and F. Rosello Llompart (eds.), Proc. Colloquium on Graph Transformation and its Application in Computer Science, Technical Report B-19, Universitat de les Illes Balears, 1995.

  36. Meseguer, J. and Montanari, U.: Petri nets are monoids, Inform. and Comput. 88(2) (1990), 105-155.

    Google Scholar 

  37. Meseguer, J.: Conditional rewriting logic as a unified model of concurrency, TCS 96 (1992), 73-155.

    Google Scholar 

  38. Reichel, H.: An approach to object semantics based on terminal co-algebras, Math. Structures in Comput. Sci. 5 (1995), 129-152.

    Google Scholar 

  39. Rutten, J.: Universal coalgebra: A theory of systems, Technical Report CS-R9652, CWI, 1996. To appear in TCS.

  40. Jacobs, B.: Inheritance and cofree constructions, in P. Cointe (ed.), European Conference on Object-Oriented Programming (ECOOP), Springer LNCS 1098, Berlin, 1996.

    Google Scholar 

  41. Jacobs, B. and Rutten, J.: A tutorial on (co)algebras and (co)induction, Bulletin of EATCS 62 (1997), 222-259.

    Google Scholar 

  42. Turi, D. and Plotkin, G.: Towards a mathematical operational semantics, in Proc. of LICS'97, 1997, pp. 280-305.

  43. Jacobs, B., Moss, L., Reichel, H. and Rutten, J. (eds.): 1thWorkshop on Coalgebraic Methods in Computer Science (CMCS'98), Lisbon, Portugal, Electronic Notes of TCS 11, Elsevier Science, 1998. http://www.elsevier.nl/locate/entcs.

  44. Barr, M.: Terminal coalgebras in well-founded set theory, Theoret. Comput. Sci. 114 (1993), 299-315.

    Google Scholar 

  45. Habel, A., Heckel, R. and Taentzer, G.: Graph grammars with negative application conditions, Fund. Inform. 26(3-4) (1996), 287-313.

    Google Scholar 

  46. Heckel, R. and Wagner, A.: Ensuring consistency of conditional graph grammars-a constructive approach, in Proc. of SEGRAGRA'95 "Graph Rewriting and Computation", Electronic Notes of TCS 2, 1995. http://www.elsevier.nl/locate/entcs/volume2.html.

  47. Heckel, R., Ehrig, H., Wolter, U. and Corradini, A.: Integrating the specification techniques of graph transformation and temporal logic, in Proc. Mathematical Foundations of Computer Science (MFCS'97), Bratislava, LNCS 1295, Springer-Verlag, 1997, pp. 219-228.

  48. Gadducci, F., Heckel, R. and Koch, M.: Model checking graph-interpreted temporal formulas, in Prelim. Proc. 6th Int. Workshop on Theory and Application of Graph Transformation (TAGT'98), Paderborn, 1998.

  49. Corradini, A.: Concurrent graph and term graph rewriting, in U. Montanari and V. Sassone (eds.), Proc. CONCUR'96, LNCS 1119, Springer-Verlag, 1996, pp. 438-464.

  50. Nielsen, M., Priese, L. and Sassone, V.: Characterizing behavioural congruences for Petri nets, in Proc. CONCUR'95, LNCS 962, Springer-Verlag, 1995, pp. 175-189.

  51. Ehrig, H., Merten, A. and Padberg, J.: How to transfer concepts of abstract data types to petri nets?, EATCS Bulletin 62 (1997), 106-114.

    Google Scholar 

  52. Padberg, J., Jansen, L., Heckel, R. and Ehrig, H.: Interoperability in train control systems: Specification of scenarios using open nets, in Proc. Integrated Design and Process Technology (IDPT'98), Berlin, 1998.

  53. Ehrig, H., Heckel, R., Padberg, J. and Rozenberg, G.: Graph transformation and other rulebased formalisms with incomplete information, in Prelim. Proc. 6th Int. Workshop on Theory and Application of Graph Transformation (TAGT'98), Paderborn, 1998.

  54. Corradini, A., Große-Rhode, M. and Heckel, R.: Structured transition systems as lax coalgebras, in B. Jacobs et al. [43].

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Heckel, R., Ehrig, H., Wolter, U. et al. Double-Pullback Transitions and Coalgebraic Loose Semantics for Graph Transformation Systems. Applied Categorical Structures 9, 83–110 (2001). https://doi.org/10.1023/A:1008734426504

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