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How Petri Net Theory Serves Petri Net Model Checking: A Survey

  • Karsten WolfEmail author
Chapter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11790)

Abstract

Structure theory is a unique treasure of the Petri net community. It was originally studied as a set of stand-alone techniques for exploring Petri net properties such as liveness, boundedness, reachability, and deadlock freedom. Today, methods based on the exploration of the reachability graph (state space methods) dominate Petri net verification. Thanks to the concept of model checking, these methods can deal with a much larger range of verification problems, and thanks to state space reduction methods (symmetries, partial order reduction, and other abstraction techniques), they became tractable for many practical applications. However, in the course of pushing model checking technology to its limits, several elements of Petri net structure theory celebrate a resurrection, being viewed from a different angle. This time, they are used for acceleration of the state space methods. In this article, we give an overview on the use of structural methods in Petri net model checking. We further report on our experience with combining state space and structural methods.

Keywords

Model checking State equation Petri net invariant Siphons and traps Conflict cluster 

References

  1. 1.
    Behrmann, G., Larsen, K.G., Pelánek, R.: To store or not to store. In: Hunt, W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 433–445. Springer, Heidelberg (2003).  https://doi.org/10.1007/978-3-540-45069-6_40CrossRefGoogle Scholar
  2. 2.
    Bønneland, F., Dyhr, J., Jensen, P.G., Johannsen, M., Srba, J.: Simplification of CTL formulae for efficient model checking of Petri nets. In: Khomenko, V., Roux, O.H. (eds.) PETRI NETS 2018. LNCS, vol. 10877, pp. 143–163. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-91268-4_8CrossRefGoogle Scholar
  3. 3.
    Bryant, R.E.: Symbolic Boolean manipulation with ordered binary-decision diagrams. ACM Comput. Surv. 24(3), 293–318 (1992)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Büchi, J.R.: On a decision method in restricted second order arithmetic. In: Proceedings of the International Congress on Logic, Method, and Philosophy of Science, pp. 1–12 (1962)Google Scholar
  5. 5.
    Burch, J.R., Clarke, E.M., McMillan, K.L., Dill, D.L., Hwang, L.J.: Symbolic model checking: 10\(^{20}\) states and beyond. In: Proceedings of the LICS, pp. 428–439. IEEE (1990)Google Scholar
  6. 6.
    Christensen, S., Kristensen, L.M., Mailund, T.: A sweep-line method for state space exploration. In: Margaria, T., Yi, W. (eds.) TACAS 2001. LNCS, vol. 2031, pp. 450–464. Springer, Heidelberg (2001).  https://doi.org/10.1007/3-540-45319-9_31CrossRefzbMATHGoogle Scholar
  7. 7.
    Clarke, E.M., Biere, A., Raimi, R., Zhu, Y.: Bounded model checking using satisfiability solving. Formal Methods Syst. Des. 19(1), 7–34 (2001)CrossRefGoogle Scholar
  8. 8.
    Clarke, E.M., Emerson, E.A., Sistla, A.P.: Automatic verification of finite-state concurrent systems using temporal logic specifications. ACM Trans. Program. Lang. Syst. 8(2), 244–263 (1986)CrossRefGoogle Scholar
  9. 9.
    Commoner, F., Holt, A.W., Even, S., Pnueli, A.: Marked directed graphs. J. Comput. Syst. Sci. 5(5), 511–523 (1971)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cook, S.A.: The complexity of theorem-proving procedures. In: Proceedings of the 3rd Annual ACM Symposium on Theory of Computing, pp. 151–158 (1971)Google Scholar
  11. 11.
    Emerson, E.A., Clarke, E.M.: Using branching time temporal logic to synthesize synchronization skeletons. Sci. Comput. Program. 2(3), 241–266 (1982)CrossRefGoogle Scholar
  12. 12.
    Esparza, J.: Model checking using net unfoldings. In: Gaudel, M.-C., Jouannaud, J.-P. (eds.) CAAP 1993. LNCS, vol. 668, pp. 613–628. Springer, Heidelberg (1993).  https://doi.org/10.1007/3-540-56610-4_93CrossRefGoogle Scholar
  13. 13.
    Esparza, J., Melzer, S.: Model checking LTL using constraint programming. In: Azéma, P., Balbo, G. (eds.) ICATPN 1997. LNCS, vol. 1248, pp. 1–20. Springer, Heidelberg (1997).  https://doi.org/10.1007/3-540-63139-9_26CrossRefGoogle Scholar
  14. 14.
    Evangelista, S., Kristensen, L.M.: A sweep-line method for Büchi automata-based model checking. Fundam. Inform. 131(1), 27–53 (2014)zbMATHGoogle Scholar
  15. 15.
    Garavel, H.: Nested-unit Petri nets: a structural means to increase efficiency and scalability of verification on elementary nets. In: Devillers, R., Valmari, A. (eds.) PETRI NETS 2015. LNCS, vol. 9115, pp. 179–199. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-19488-2_9CrossRefzbMATHGoogle Scholar
  16. 16.
    Godefroid, P., Wolper, P.: A partial approach to model checking. Inf. Comput. 110(2), 305–326 (1994)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hack, M.: Analysis of production schemata by Petri nets. Technical report, MS thesis, Department of Electrical Engineering, MIT, Cambridge, Massachusetts (1972)Google Scholar
  18. 18.
    Hack, M.: Decidability questions for Petri nets. Outstanding Dissertations in the Computer Sciences. Garland Publishing, New York (1975)Google Scholar
  19. 19.
    Hajdu, Á., Vörös, A., Bartha, T.: New search strategies for the Petri net CEGAR approach. In: Devillers, R., Valmari, A. (eds.) PETRI NETS 2015. LNCS, vol. 9115, pp. 309–328. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-19488-2_16CrossRefzbMATHGoogle Scholar
  20. 20.
    Heiner, M., Rohr, C., Schwarick, M., Tovchigrechko, A.A.: MARCIE’s secrets of efficient model checking. In: Koutny, M., Desel, J., Kleijn, J. (eds.) Transactions on Petri Nets and Other Models of Concurrency XI. LNCS, vol. 9930, pp. 286–296. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53401-4_14CrossRefGoogle Scholar
  21. 21.
    Holt, A.W., Commoner, F.: Events and conditions. In: MAC Conference on Concurrent Systems and Parallel Computation, pp. 3–52 (1970)Google Scholar
  22. 22.
    Jensen, K.: Condensed state spaces for symmetrical coloured Petri nets. Formal Methods Syst. Des. 9(1/2), 7–40 (1996)CrossRefGoogle Scholar
  23. 23.
    Kam, T., Villa, T., Brayton, R.K.: Multi-valued decision diagrams: theory and applications. Multiple-Valued Logic 4(01), 9–62 (1998)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Karp, R.M., Miller, R.E.: Parallel program schemata. J. Comput. Syst. Sci. 3(2), 147–195 (1969)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Kordon, F., et al.: MCC’2015 – the fifth model checking contest. In: Koutny, M., Desel, J., Kleijn, J. (eds.) Transactions on Petri Nets and Other Models of Concurrency XI. LNCS, vol. 9930, pp. 262–273. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53401-4_12CrossRefGoogle Scholar
  26. 26.
    Kosaraju, S.R.: Decidability of reachability on vector addition systems. In: ACM Symposium on Theory Computing, pp. 267–281 (1982)Google Scholar
  27. 27.
    Kristensen, L.M., Mailund, T.: A generalised sweep-line method for safety properties. In: Eriksson, L.-H., Lindsay, P.A. (eds.) FME 2002. LNCS, vol. 2391, pp. 549–567. Springer, Heidelberg (2002).  https://doi.org/10.1007/3-540-45614-7_31CrossRefGoogle Scholar
  28. 28.
    Lautenbach, K., Schmid, H.A.: Use of Petri nets for proving correctness of concurrent process systems. In: IFIP Congress, pp. 187–191 (1974)Google Scholar
  29. 29.
    Leroux, J.: The general vector addition system reachability problem by Presburger inductive invariants. In: Proceedings of the LICS, pp. 4–13. IEEE (2009)Google Scholar
  30. 30.
    Lipton, R.J.: The reachability problem requires exponential space. Technical report 62, Department of Computer Science, Yale University (1976)Google Scholar
  31. 31.
    Manna, Z., Pnueli, A.: The Temporal Logic of Reactive and Concurrent Systems - Specification. Springer, New York (1992).  https://doi.org/10.1007/978-1-4612-0931-7CrossRefzbMATHGoogle Scholar
  32. 32.
    Mayr, E.W.: An algroithm for the general Petri net reachability problem. SIAM J. Comput. 13(3), 441–460 (1984)MathSciNetCrossRefGoogle Scholar
  33. 33.
    McMillan, K.L.: Using unfoldings to avoid the state explosion problem in the verification of asynchronous circuits. In: von Bochmann, G., Probst, D.K. (eds.) CAV 1992. LNCS, vol. 663, pp. 164–177. Springer, Heidelberg (1993).  https://doi.org/10.1007/3-540-56496-9_14CrossRefGoogle Scholar
  34. 34.
    Memmi, G., Roucairol, G.: Linear algebra in net theory. In: Brauer, W. (ed.) Net Theory and Applications. LNCS, vol. 84, pp. 213–223. Springer, Heidelberg (1980).  https://doi.org/10.1007/3-540-10001-6_24CrossRefGoogle Scholar
  35. 35.
    Miner, A.S., Ciardo, G.: Efficient reachability set generation and storage using decision diagrams. In: Donatelli, S., Kleijn, J. (eds.) ICATPN 1999. LNCS, vol. 1639, pp. 6–25. Springer, Heidelberg (1999).  https://doi.org/10.1007/3-540-48745-X_2CrossRefGoogle Scholar
  36. 36.
    Oanea, O., Wimmel, H., Wolf, K.: New algorithms for deciding the siphon-trap property. In: Lilius, J., Penczek, W. (eds.) PETRI NETS 2010. LNCS, vol. 6128, pp. 267–286. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-13675-7_16CrossRefGoogle Scholar
  37. 37.
    Pastor, E., Cortadella, J., Peña, M.A.: Structural methods to improve the symbolic analysis of Petri nets. In: Donatelli, S., Kleijn, J. (eds.) ICATPN 1999. LNCS, vol. 1639, pp. 26–45. Springer, Heidelberg (1999).  https://doi.org/10.1007/3-540-48745-X_3CrossRefGoogle Scholar
  38. 38.
    Pastor, E., Roig, O., Cortadella, J., Badia, R.M.: Petri net analysis using boolean manipulation. In: Valette, R. (ed.) ICATPN 1994. LNCS, vol. 815, pp. 416–435. Springer, Heidelberg (1994).  https://doi.org/10.1007/3-540-58152-9_23CrossRefGoogle Scholar
  39. 39.
    Peled, D.: All from one, one for all: on model checking using representatives. In: Courcoubetis, C. (ed.) CAV 1993. LNCS, vol. 697, pp. 409–423. Springer, Heidelberg (1993).  https://doi.org/10.1007/3-540-56922-7_34CrossRefGoogle Scholar
  40. 40.
    Ridder, H.: Analyse von Petri-Netz-Modellen mit Entscheidungsdiagrammen. Ph.D. thesis, Koblenz, Landau, University (1997)Google Scholar
  41. 41.
    Roch, S., Starke, P.H.: INA: integrated net analyzer (1999)Google Scholar
  42. 42.
    Schmidt, K.: Stubborn sets for standard properties. In: Donatelli, S., Kleijn, J. (eds.) ICATPN 1999. LNCS, vol. 1639, pp. 46–65. Springer, Heidelberg (1999).  https://doi.org/10.1007/3-540-48745-X_4CrossRefGoogle Scholar
  43. 43.
    Schmidt, K.: How to calculate symmetries of Petri nets. Acta Inf. 36(7), 545–590 (2000)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Schmidt, K.: Using Petri net invariants in state space construction. In: Garavel, H., Hatcliff, J. (eds.) TACAS 2003. LNCS, vol. 2619, pp. 473–488. Springer, Heidelberg (2003).  https://doi.org/10.1007/3-540-36577-X_35CrossRefGoogle Scholar
  45. 45.
    Schmidt, K.: Automated generation of a progress measure for the sweep-line method. STTT 8(3), 195–203 (2006)CrossRefGoogle Scholar
  46. 46.
    Starke, P.H.: Reachability analysis of Petri nets using symmetries. Syst. Anal. Model. Simul. 8(4–5), 293–303 (1991)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Strehl, K., Thiele, L.: Interval diagram techniques for symbolic model checking of Petri nets. In: Proceedings of the Design, Automation and Test in Europe, pp. 756–757 (1999)Google Scholar
  48. 48.
    Tarjan, R.E.: Depth-first search and linear graph algorithms. SIAM J. Comput. 1(2), 146–160 (1972)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Tarjan, R.E.: Efficiency of a good but not linear set union algorithm. J. ACM 22(2), 215–225 (1975)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Triebel, M., Sürmeli, J.: Characterizing stable and deriving valid inequalities of Petri nets. Fundam. Inform. 146(1), 1–34 (2016)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Valmari, A.: Stubborn sets for reduced state space generation. In: Rozenberg, G. (ed.) ICATPN 1989. LNCS, vol. 483, pp. 491–515. Springer, Heidelberg (1991).  https://doi.org/10.1007/3-540-53863-1_36CrossRefGoogle Scholar
  52. 52.
    Valmari, A.: The state explosion problem. In: Reisig, W., Rozenberg, G. (eds.) ACPN 1996. LNCS, vol. 1491, pp. 429–528. Springer, Heidelberg (1998).  https://doi.org/10.1007/3-540-65306-6_21CrossRefGoogle Scholar
  53. 53.
    Valmari, A., Hansen, H.: Stubborn set intuition explained. In: Koutny, M., Kleijn, J., Penczek, W. (eds.) Transactions on Petri Nets and Other Models of Concurrency XII. LNCS, vol. 10470, pp. 140–165. Springer, Heidelberg (2017).  https://doi.org/10.1007/978-3-662-55862-1_7CrossRefGoogle Scholar
  54. 54.
    Vardi, M.Y.: Verification of concurrent programs: the automata-theoretic framework. In: Proceedings of the LICS, pp. 167–176. IEEE (1987)Google Scholar
  55. 55.
    Vergauwen, B., Lewi, J.: A linear local model checking algorithm for CTL. In: Best, E. (ed.) CONCUR 1993. LNCS, vol. 715, pp. 447–461. Springer, Heidelberg (1993).  https://doi.org/10.1007/3-540-57208-2_31CrossRefGoogle Scholar
  56. 56.
    Wimmel, H., Wolf, K.: Applying CEGAR to the Petri net state equation. Log. Methods Comput. Sci. 8(3) (2012)Google Scholar
  57. 57.
    Wolf, K.: Petri net model checking with LoLA 2. In: Khomenko, V., Roux, O.H. (eds.) PETRI NETS 2018. LNCS, vol. 10877, pp. 351–362. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-91268-4_18CrossRefGoogle Scholar
  58. 58.
    Wolf, K.: A simple abstract interpretation for Petri net queries. In: Proceedings of the PNSE, CEUR Workshop Proceedings, vol. 2138, pp. 163–170 (2018)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut für InformatikUniversität RostockRostockGermany

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