Skip to main content

Firing Partial Orders in a Petri Net

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 12734)

Abstract

Petri nets have the simple firing rule that a transition is enabled to fire if its preset of places is marked. The occurrence of a transition is called an event. To check whether a sequence of events is enabled, we simply try to fire the sequence from ‘start’ to ‘end’ in the initial marking of the net. It is a bit of a stretch to call this an algorithm, but its runtime complexity is in \(O(|P| \cdot |V|)\), where P is the set of places and V is the set of events.

Petri nets model distributed systems. An execution of a distributed system is a partial order of events rather than a sequence. Compact tokenflows are tailored to an efficient algorithm that decides if a partial order of events is enabled in a Petri net. Yet, the runtime complexity of this algorithm is in \(O(|P| \cdot |V|^3)\).

In practical applications dealing with a huge amount of behavioral data, the gap between just firing a sequence and deciding if a partial order is enabled, makes a big difference.

In this paper, we present an approach to just firing a partial order of events in a Petri net. By firing a partial order, we obtain a lot of information about whether or not the partial order is enabled. We show that just firing is often enough if done correctly.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-76983-3_20
  • Chapter length: 21 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   84.99
Price excludes VAT (USA)
  • ISBN: 978-3-030-76983-3
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   109.99
Price excludes VAT (USA)
Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
Fig. 7.
Fig. 8.
Fig. 9.
Fig. 10.
Fig. 11.
Fig. 12.
Fig. 13.
Fig. 14.

References

  1. van der Aalst, W.M.P., van Dongen, B.F.: Discovering petri nets from event logs. In: Jensen, K., van der Aalst, W.M.P., Balbo, G., Koutny, M., Wolf, K. (eds.) Transactions on Petri Nets and Other Models of Concurrency VII. LNCS, vol. 7480, pp. 372–422. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38143-0_10

    CrossRef  Google Scholar 

  2. van der Aalst, W.M.P.: The application of petri nets to workflow management. J. Circ. Syst. Comput. 8(1), 21–66 (1998)

    CrossRef  Google Scholar 

  3. Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice Hall, Englewood Cliffs (1993)

    MATH  Google Scholar 

  4. Bergenthum, R., Lorenz, R.: Verification of scenarios in petri nets using compact tokenflows. Fundamenta Informaticae 137, 117–142 (2015)

    MathSciNet  CrossRef  Google Scholar 

  5. Bergenthum, R.: Faster verification of partially ordered runs in petri nets using compact tokenflows. In: Colom, J.-M., Desel, J. (eds.) PETRI NETS 2013. LNCS, vol. 7927, pp. 330–348. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38697-8_18

    CrossRef  MATH  Google Scholar 

  6. Bergenthum, R., Lorenz, R., Mauser, S.: Faster unfolding of general petri nets based on token flows. In: van Hee, K.M., Valk, R. (eds.) PETRI NETS 2008. LNCS, vol. 5062, pp. 13–32. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-68746-7_6

    CrossRef  MATH  Google Scholar 

  7. Desel, J., Juhás, G., Lorenz, R., Neumair, C.: Modelling and validation with VipTool. Bus. Process Manag. 2003, 380–389 (2003)

    CrossRef  Google Scholar 

  8. Desel, J., Juhás, G.: “What is a petri net?” informal answers for the informed reader. In: Ehrig, H., Padberg, J., Juhás, G., Rozenberg, G. (eds.) Unifying Petri Nets. LNCS, vol. 2128, pp. 1–25. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45541-8_1

    CrossRef  MATH  Google Scholar 

  9. Desel, J., Reisig, W.: Place/transition petri nets. In: Reisig, W., Rozenberg, G. (eds.) ACPN 1996. LNCS, vol. 1491, pp. 122–173. Springer, Heidelberg (1998). https://doi.org/10.1007/3-540-65306-6_15

    CrossRef  MATH  Google Scholar 

  10. Dumas, M., García-Bañuelos, L.: Process mining reloaded: event structures as a unified representation of process models and event logs. In: Devillers, R., Valmari, A. (eds.) PETRI NETS 2015. LNCS, vol. 9115, pp. 33–48. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-19488-2_2

    CrossRef  MATH  Google Scholar 

  11. Desel, J., Erwin, T.: Quantitative Engineering of Business Processes with VIPbusiness. In: Ehrig, H., Reisig, W., Rozenberg, G., Weber, H. (eds.) Petri Net Technology for Communication-Based Systems. LNCS, vol. 2472, pp. 219–242. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-40022-6_11

    CrossRef  Google Scholar 

  12. Fahland, D.: Scenario-based process modeling with Greta. BPM Demonstration Track 2010, CEUR 615 (2010)

    Google Scholar 

  13. Fahland, D.: Oclets – scenario-based modeling with petri nets. In: Franceschinis, G., Wolf, K. (eds.) PETRI NETS 2009. LNCS, vol. 5606, pp. 223–242. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02424-5_14

    CrossRef  MATH  Google Scholar 

  14. Ford, L.R., Fulkerson, D.R.: Maximal flow through a network. Can. J. Math. 8, 399–404 (1956)

    MathSciNet  CrossRef  Google Scholar 

  15. Grabowski, J.: On partial languages. Fundamenta Informaticae 4, 427–498 (1981)

    MathSciNet  CrossRef  Google Scholar 

  16. Goltz, U., Reisig, W.: Processes of place/transition-nets. In: Diaz, J. (ed.) ICALP 1983. LNCS, vol. 154, pp. 264–277. Springer, Heidelberg (1983). https://doi.org/10.1007/BFb0036914

    CrossRef  Google Scholar 

  17. Juhás, G., Lorenz, R., Desel, J.: Can i execute my scenario in your net? In: Ciardo, G., Darondeau, P. (eds.) ICATPN 2005. LNCS, vol. 3536, pp. 289–308. Springer, Heidelberg (2005). https://doi.org/10.1007/11494744_17

    CrossRef  Google Scholar 

  18. Karzanov, A.: Determining the maximal flow in a network by the method of preflows. Doklady Math. 15, 434–437 (1974)

    MATH  Google Scholar 

  19. Kiehn, A.: On the interrelation between synchronized and non-synchronized behaviour of petri nets. Elektronische Informationsverarbeitung und Kybernetik 2(1/2), 3–18 (1988)

    MathSciNet  MATH  Google Scholar 

  20. Mannel, L.L., van der Aalst, W.M.P.: Finding complex process-structures by exploiting the token-game. In: Donatelli, S., Haar, S. (eds.) PETRI NETS 2019. LNCS, vol. 11522, pp. 258–278. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-21571-2_15

    CrossRef  Google Scholar 

  21. Peterson, J.L.: Petri Net Theory and the Modeling of Systems. Prentice-Hall, Englewood Cliffs (1981)

    MATH  Google Scholar 

  22. Reisig, W.: Understanding Petri Nets - Modeling Techniques, Analysis Methods, Case Studies. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-33278-4

    CrossRef  MATH  Google Scholar 

  23. Vogler, W. (ed.): Modular Construction and Partial Order Semantics of Petri Nets. LNCS, vol. 625. Springer, Heidelberg (1992). https://doi.org/10.1007/3-540-55767-9

    CrossRef  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Robin Bergenthum .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Bergenthum, R. (2021). Firing Partial Orders in a Petri Net. In: Buchs, D., Carmona, J. (eds) Application and Theory of Petri Nets and Concurrency. PETRI NETS 2021. Lecture Notes in Computer Science(), vol 12734. Springer, Cham. https://doi.org/10.1007/978-3-030-76983-3_20

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-76983-3_20

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-76982-6

  • Online ISBN: 978-3-030-76983-3

  • eBook Packages: Computer ScienceComputer Science (R0)