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Hierarchical Reachability Graph Generation for Petri Nets

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Abstract

Reachability analysis is the most general approach to the analysis of Petri nets. Due to the well-known problem of state-space explosion, generation of the reachability set and reachability graph with the known approaches often becomes intractable even for moderately sized nets. This paper presents a new method to generate and represent the reachability set and reachability graph of large Petri nets in a compositional and hierarchical way. The representation is related to previously known Kronecker-based representations, and contains the complete information about reachable markings and possible transitions. Consequently, all properties that it is possible for the reachability graph to decide can be decided using the Kronecker representation. The central idea of the new technique is a divide and conquer approach. Based on net-level results, nets are decomposed, and reachability graphs for parts are generated and combined. The whole approach can be realized in a completely automated way and has been integrated in a Petri net-based analysis tool.

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References

  1. S. Allmaier, M. Kowarschik, and G. Horton, “State space construction and steady state solution of GSPNs on a shared-memory multiprocessor,” in Proc. 7th Int. Workshop on Petri Nets and Performance Models (PNPM'97), IEEE CS Press, 1997, pp. 112-121.

  2. F. Bause, P. Kemper, and P. Kritzinger, “Abstract Petri net notation,” Petri Net Newsletter, Vol. 49, pp. 9-27, 1995.

    Google Scholar 

  3. G. Berthelot, “Transformation and decomposition of nets,” in G. Rozenberg (Ed.), Advances in Petri Nets 85, Springer, 1986. LNCS, Vol. 254.

  4. P. Buchholz, “Hierarchical high level Petri nets for complex system analysis,” in R. Valette (Ed.), Application and Theory of Petri Nets, Springer, 1994, pp. 119-138. LNCS, Vol. 815.

  5. P. Buchholz, “Markovian process algebra: Composition and equivalence,” in U. Herzog and M. Rettelbach (Eds.), Proc. 2nd Workshop on Process Algebras and Performance Modelling, 1994; Arbeitsberichte des IMMD, Vol. 27, No. 4, pp. 11-30, 1994.

  6. P. Buchholz, “Hierarchical structuring of superposed GSPNs,” IEEE Trans. on Softw. Eng., Vol. 25, No. 2, pp. 81-90, 1999.

    Google Scholar 

  7. P. Buchholz, “Structured analysis approaches for large Markov chains,” Applied Numerical Mathematics, Vol. 31, No. 4, pp. 375-404, 1999.

    Google Scholar 

  8. P. Buchholz and P. Kemper, “Hierarchical reachability graph generation for Petri nets,” Forschungsbericht Nr. 660 des Fachbereichs Informatik der Universität Dortmund (Germany), 1997.

  9. P. Buchholz and P. Kemper, “On generating a hierarchy for GSPN analysis,” ACM Performance Evaluation Review, Vol. 26, No. 2, pp. 5-14, 1998.

    Google Scholar 

  10. P. Buchholz and P. Kemper, “A toolbox for the analysis of discrete event dynamic systems,” in N. Halbwachs and D. Peled (Eds.), Computer Aided Verification (CAV'99), Springer, 1999, pp. 483-486, LNCS, Vol. 1633.

  11. P. Buchholz and P. Kemper, “Efficient computation and representation of large reachability sets for composed automata,” in R. Boel and G. Stremersch (Eds.), Discrete Event Systems Analysis and Control, Kluwer Academic, 2000, pp. 49-56.

  12. J.R. Burch, E.M. Clarke, K.L. McMillan, D.L. Dill, and J. Hwang, “Symbolic model checking: 1020 states and beyond,” in Proc. 5th Annual Symposium on Logic in Computer Science, June 1990.

  13. J. Campos, S. Donatelli, and J. Silva, “Structured solution of asynchronously communicating stochastic modules,” IEEE Trans. on Softw. Engineering, Vol. 25, No. 2, pp. 147-165, 1999.

    Google Scholar 

  14. S. Caselli, G. Conte, and P. Marenzoni, “Parallel state space exploration for GSPN models,” in G. De Michelis and M. Diaz (Eds.), Application and Theory of Petri Nets 1995, Springer, 1995, pp. 181-200, LNCS, Vol. 935.

  15. G. Chiola, C. Dutheillet, G. Franceschini, and S. Haddad, “A symbolic reachability graph for coloured Petri nets,” Theoretical Computer Science, Vol. 176, pp. 39-65, 1997.

    Google Scholar 

  16. G. Chiola and A. Ferscha, “Distributed simulation of Petri nets,” IEEE Parallel and Distributed Technology, Vol. 1, pp. 33-50, 1993.

    Google Scholar 

  17. S. Christensen and L. Petrucci, “Modular state space analysis of coloured Petri nets,” in G. De Michelis and M. Diaz (Eds.), Application and Theory of Petri Nets, Springer, 1995, pp. 201-217, LNCS, Vol. 935.

  18. G. Ciardo, G. Luettgen, and R. Siminiceanu, “Saturation: An efficient iteration strategy for symbolic statespace generation,” in T. Margaria and W. Yi (Eds.), Tools and Algorithms for the Construction and Analysis of Systems (TACAS 2001), Springer, 2001, pp. 328-342, LNCS, Vol. 2031.

  19. G. Ciardo and A.S. Miner, “Storage alternatives for large structured state spaces,” in R. Marie, B. Plateau, M. Calzarossa, and G. Rubino (Eds.), Proc. 9th Int. Conf. Modelling Techniques and Tools for Computer Performance Evaluation, St. Malo, France, 1997, Springer, 1997, pp. 44-57. LNCS, Vol. 1245.

  20. E.M. Clarke, E.A. Emerson, and A.P. Sistla, “Automatic verification of finite-state concurrent systems using temporal logic specifications,” ACM Transactions on Programming Languages and Systems, Vol. 8, No. 2, pp. 244-263, 1986.

    Google Scholar 

  21. R. Cleaveland, J. Parrow, and B. Steffen, “The concurrency workbench: A semantics based tool for the verification of concurrent systems,” ACM Transactions on Programming Languages and Systems, Vol. 15, No. 1, pp. 36-72, 1993.

    Google Scholar 

  22. M. Davio, “Kronecker products and shuffle algebra,” IEEE Trans. on Comp., Vol. 30, pp. 116-125, 1981.

    Google Scholar 

  23. S. Donatelli, “Superposed generalized stochastic Petri nets: Definition and efficient solution,” in R. Valette (Ed.), Application and Theory of Petri Nets, Springer, 1994, LNCS, Vol. 815.

  24. P. Godefroid and P. Wolper, “Using partial orders for the efficient verification of deadlock freedom and safety properties,” Formal Methods in System Design, Vol. 2, pp. 149-164, 1993.

    Google Scholar 

  25. S. Haddad, “A reduction theory for coloured Petri nets,” in G. Rozenberg (Ed.), Advances in Petri Nets 1989, Springer, 1990, pp. 209-235. LNCS, Vol. 424.

  26. M. Heiner and P. Deusen, “Petri net based design and analysis of reactive systems,” in Proc. 3rd Workshop on Discrete Event Systems (WoDES'96), pp. 308-313.

  27. M. Heiner, P. Deussen, and J. Spranger, “A case study in developing control software on manufacturing systems,” in M. Silva, R. Valette, and K. Takahashi (Eds.), Workshop on Manufacturing and Petri nets, within 17th Int. Conf. Application and Theory of Petri Nets, Osaka, Japan, 1996.

  28. G.J. Holzmann, “On limits and possibilities of automated protocol analysis,” in Proc. 7th Int. Workshop Protocol Specification, Testing and Verification, North Holland, 1987.

  29. G.J. Holzmann, “An analysis of bitstate hashing,” in Proc. 15th Int. Symp. Protocol Specification, Testing and Verification, IFIP, Chapman & Hall, 1995.

  30. K. Jensen, “Coloured Petri nets and the invariant method,” Theor. Comp. Sci., Vol. 14, pp. 317-336, 1981.

    Google Scholar 

  31. K. Jensen, “Coloured Petri nets,” Vol. 1; Springer, EATCS Monographs (1992).

  32. P. Kemper, “Reachability analysis based on structured representations,” in J. Billington and W. Reisig (Eds.), Application and Theory of Petri Nets 1996. Springer, 1996, pp. 269-288. LNCS, Vol. 1091.

  33. C. Lewerentz and T. Lindner (Eds.), “Formal development reactive systems,” Springer, 1995, LNCS, Vol. 891.

  34. P. Marenzoni, S. Caselli, and G. Conte, “Analysis of large GSPN models: A distributed solution tool,” in. Proc. 7th Int. Workshop on Petri Nets and Performance Models (PNPM'97), IEEE CS Press, 1997, pp. 122-131.

  35. J. Martinez and M. Silva, “A simple and fast algorithm to obtain all invariants of a generalized Petri net,” in Application and Theory of Petri Nets 1981, Springer, IFB 52, 1981.

  36. R. Milner, “Communication and concurrency,” Prentice Hall, 1989.

  37. T. Murata, “Petri nets: Properties, analysis and applications,” Proc. of the IEEE, Vol. 77, pp. 541-580, 1989.

    Google Scholar 

  38. D. Nicol and G. Ciardo, “Automated parallelization of discrete state-space generation,” Journal of Parallel and Distributed Computing, Vol. 47, pp. 153-167, 1997.

    Google Scholar 

  39. M. Notomi and T. Murata, “Hierarchical reachability graph generation of bounded Petri nets for concurrentsoftware analysis,” IEEE Trans. on Softw. Eng., Vol. 20, pp. 325-336, 1994.

    Google Scholar 

  40. E. Pastor, O. Roig, J. Cortadella, and R.M. Badia, “Petri net analysis using Boolean manipulation,” in R. Valette (Ed.), Application and Theory of Petri Nets 1994, Springer, 1994. LNCS, Vol. 815.

  41. L. Pomello, G. Rozenberg, and C. Simone, “A survey of equivalence relations for net based systems,” in G. Rozenberg (Ed.), Advances in Petri Nets 1992, Springer, 1992, pp. 410-472. LNCS, Vol. 609.

  42. Proc. 7th Int. Workshop on Petri Nets and Performance Models (PNPM'97), IEEE CS Press, 1997.

  43. L. Recalde, E. Teruel, and M. Silva; {SC}* ECS: “A class of modular and hierarchical cooperating systems.” in J. Billington and W. Reisig (Eds.), Application and Theory of Petri Nets 1996, Springer, 1996, pp. 440-459. LNCS, Vol. 1091.

  44. R. Valette (Ed.), Application and Theory of Petri Nets, Springer, 1994. LNCS, Vol. 815.

  45. A. Valmari, “Compositional analysis with place bordered subnets,” in R. Valette (Ed.), Application and Theory of Petri Nets, Springer, 1994, pp. 531-547. LNCS, Vol. 815.

  46. A. Valmari, “State of the art report: Stubborn sets,” Petri Net Newsletter, Vol. 46, pp. 6-14, 1994.

    Google Scholar 

  47. P. Wolper and D. Leroy, “Reliable hashing without collision detection,” in 5th Int. Conf. Computer Aided Verification, Elounda, Greece, 1993.

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Authors and Affiliations

  1. Fakultät Informatik, TU Dresden, D-01062, Dresden, Germany

    Peter Buchholz

  2. Informatik IV, Universität Dortmund, D-44221, Dortmund, Germany

    Peter Kemper

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  1. Peter Buchholz
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  2. Peter Kemper
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Buchholz, P., Kemper, P. Hierarchical Reachability Graph Generation for Petri Nets. Formal Methods in System Design 21, 281–315 (2002). https://doi.org/10.1023/A:1020321222420

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