A Boolean Model for Enumerating Minimal Siphons and Traps in Petri Nets

  • Faten Nabli
  • François Fages
  • Thierry Martinez
  • Sylvain Soliman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7514)


Petri nets are a simple formalism for modeling concurrent computation. Recently, they have emerged as a promising tool for modeling and analyzing biochemical interaction networks, bridging the gap between purely qualitative and quantitative models. Biological networks can indeed be large and complex, which makes their study difficult and computationally challenging. In this paper, we focus on two structural properties of Petri nets, siphons and traps, that bring us information about the persistence of some molecular species. We present two methods for enumerating all minimal siphons and traps of a Petri net by iterating the resolution of Boolean satisfiability problems executed with either a SAT solver or a CLP(B) program. We compare the performances of these methods with respect to a state-of-the-art algorithm from the Petri net community. On a benchmark with 80 Petri nets from the Petriweb database and 403 Petri nets from curated biological models of the Biomodels database, we show that miniSAT and CLP(B) solvers are overall both faster by two orders of magnitude with respect to the dedicated algorithm. Furthermore, we analyse why these programs perform so well on even very large biological models and show the existence of hard instances in Petri nets with unbounded degrees.


Boolean Model Hard Instance Minimal Siphon Biomodels Database Boolean Constraint 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Birtwistle, M.R., Hatakeyama, M., Yumoto, N., Ogunnaike, B.A., Hoek, J.B., Kholodenko, B.N.: Ligand-dependent responses of the ErbB signaling network: experimental and modeling analysis. Molecular Systems Biology 3(144) (September 2007)Google Scholar
  2. 2.
    Chabrier-Rivier, N., Chiaverini, M., Danos, V., Fages, F., Schächter, V.: Modeling and querying biochemical interaction networks. Theoretical Computer Science 325(1), 25–44 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chu, F., Xie, X.-L.: Deadlock analysis of petri nets using siphons and mathematical programming. IEEE Transactions on Robotics and Automation 13(6), 793–804 (1997)CrossRefGoogle Scholar
  4. 4.
    Cordone, R., Ferrarini, L., Piroddi, L.: Characterization of minimal and basis siphons with predicate logic and binary programming. In: Proceedings of IEEE International Symposium on Computer-Aided Control System Design, pp. 193–198 (2002)Google Scholar
  5. 5.
    Cordone, R., Ferrarini, L., Piroddi, L.: Some results on the computation of minimal siphons in petri nets. In: Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, Hawaii, USA (December 2003)Google Scholar
  6. 6.
    Cordone, R., Ferrarini, L., Piroddi, L.: Enumeration algorithms for minimal siphons in petri nets based on place constraints. IEEE Transactions on Systems, Man and Cybernetics. Part A, Systems and Humans 35(6), 844–854 (2005)CrossRefGoogle Scholar
  7. 7.
    Crawford, J.M., Auton, L.D.: Experimental results on the crossover point in satisfiability problems. In: Proceedings of the 11th National Conference on Artificial Intelligence, pp. 21–27. AAAI Press (1993)Google Scholar
  8. 8.
    Diaz, D., Codognet, P.: Design and implementation of the GNU Prolog system. Journal of Functional and Logic Programming 6 (October 2001)Google Scholar
  9. 9.
    Fages, F., Soliman, S., Coolen, R.: CLPGUI: a generic graphical user interface for constraint logic programming. Journal of Constraints, Special Issue on User-Interaction in Constraint Satisfaction 9(4), 241–262 (2004)Google Scholar
  10. 10.
    Goud, R., van Hee, K.M., Post, R.D.J., van der Werf, J.M.E.M.: Petriweb: A Repository for Petri Nets. In: Donatelli, S., Thiagarajan, P.S. (eds.) ICATPN 2006. LNCS, vol. 4024, pp. 411–420. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  11. 11.
    Heiner, M., Gilbert, D., Donaldson, R.: Petri Nets for Systems and Synthetic Biology. In: Bernardo, M., Degano, P., Zavattaro, G. (eds.) SFM 2008. LNCS, vol. 5016, pp. 215–264. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  12. 12.
    Helfert, S., Estevez, A., Bakker, B., Michels, P., Clayton, C.: Roles of triosephosphate isomerase and aerobic metabolism in trypanosoma brucei. Biochem. J. 357, 117–125 (2001)CrossRefGoogle Scholar
  13. 13.
    Kinuyama, M., Murata, T.: Generating siphons and traps by petri net representation of logic equations. In: Proceedings of 2nd Conference of the Net Theory SIG-IECE, pp. 93–100 (1986)Google Scholar
  14. 14.
    Kohn, K.W.: Molecular interaction map of the mammalian cell cycle control and DNA repair systems. Molecular Biology of the Cell 10(8), 2703–2734 (1999)CrossRefGoogle Scholar
  15. 15.
    Lautenbach, K.: Linear algebraic calculation of deadlocks and traps. In: Voss, G., Rozenberg (eds.) Concurrency and Nets Advances in Petri Nets, pp. 315–336. Springer, New York (1987)CrossRefGoogle Scholar
  16. 16.
    le Novère, N., Bornstein, B., Broicher, A., Courtot, M., Donizelli, M., Dharuri, H., Li, L., Sauro, H., Schilstra, M., Shapiro, B., Snoep, J.L., Hucka, M.: BioModels Database: a free, centralized database of curated, published, quantitative kinetic models of biochemical and cellular systems. Nucleic Acid Research 1(34), D689–D691 (2006)Google Scholar
  17. 17.
    Minoux, M., Barkaoui, K.: Deadlocks and traps in petri nets as horn-satisfiability solutions and some related polynomially solvable problems. Discrete Applied Mathematics 29, 195–210 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mitchell, D., Selman, B., Levesque, H.: Hard and easy distributions of SAT problems. In: Proceedings of the 10th National Conference on Artificial Intelligence, pp. 459–465. AAAI Press (1992)Google Scholar
  19. 19.
    Murata, T.: Petri nets: properties, analysis and applications. Proceedings of the IEEE 77(4), 541–579 (1989)CrossRefGoogle Scholar
  20. 20.
    Nabli, F.: Finding minimal siphons as a CSP. In: CP 2011: The Seventeenth International Conference on Principles and Practice of Constraint Programming, Doctoral Program, pp. 67–72 (September 2011)Google Scholar
  21. 21.
    Peterson, J.L.: Petri Net Theory and the Modeling of Systems. Prentice Hall, New Jersey (1981)Google Scholar
  22. 22.
    Reddy, V.N., Mavrovouniotis, M.L., Liebman, M.N.: Petri net representations in metabolic pathways. In: Hunter, L., Searls, D.B., Shavlik, J.W. (eds.) Proceedings of the 1st International Conference on Intelligent Systems for Molecular Biology (ISMB), pp. 328–336. AAAI Press (1993)Google Scholar
  23. 23.
    Schoeberl, B., Eichler-Jonsson, C., Gilles, E., Muller, G.: Computational modeling of the dynamics of the map kinase cascade activated by surface and internalized egf receptors. Nature Biotechnology 20(4), 370–375 (2002)CrossRefGoogle Scholar
  24. 24.
    Soliman, S.: Finding minimal P/T-invariants as a CSP. In: Proceedings of the Fourth Workshop on Constraint Based Methods for Bioinformatics, WCB 2008, associated to CPAIOR 2008 (May 2008)Google Scholar
  25. 25.
    Stryer, L.: Biochemistry. Freeman, New York (1995)Google Scholar
  26. 26.
    Tanimoto, S., Yamauchi, M., Watanabe, T.: Finding minimal siphons in general petri nets. IEICE Trans. on Fundamentals in Electronics, Communications and Computer Science, 1817–1824 (1996)Google Scholar
  27. 27.
    Yamauchi, M., Watanabe, T.: Time complexity analysis of the minimal siphon extraction problem of petri nets. IEICE Trans. on Fundamentals of Electronics, Communications and Computer Sciences, 2558–2565 (1999)Google Scholar
  28. 28.
    Zevedei-Oancea, I., Schuster, S.: Topological analysis of metabolic networks based on petri net theory. Silico Biology 3(29) (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Faten Nabli
    • 1
  • François Fages
    • 1
  • Thierry Martinez
    • 1
  • Sylvain Soliman
    • 1
  1. 1.EPI ContraintesInria Paris-RocquencourtFrance

Personalised recommendations