Parametric Verification: An Introduction

  • Étienne André
  • Michał Knapik
  • Didier Lime
  • Wojciech Penczek
  • Laure PetrucciEmail author
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11790)


This paper constitutes a short introduction to parametric verification of concurrent systems. It originates from two 1-day tutorial sessions held at the Petri nets conferences in Toruń (2016) and Zaragoza (2017). A video of the presentation is available at, consisting of 14 short sequences. The paper presents not only the basic formal concepts tackled in the video version, but also an extensive literature to provide the reader with further references covering the area.

We first introduce motivation behind parametric verification in general, and then focus on different models and approaches, for verifying several kinds of systems. They include Parametric Timed Automata, for modelling real-time systems, where the timing constraints are not necessarily known a priori. Similarly, Parametric Interval Markov Chains allow for modelling systems where probabilities of events occurrences are intervals with parametric bounds. Parametric Petri Nets allow for compact representation of systems, and cope with different types of parameters. Finally, Action Synthesis aims at enabling or disabling actions in a concurrent system to guarantee some of its properties. Some tools implementing these approaches were used during hands-on sessions at the tutorial. The corresponding practicals are freely available on the Web.


  1. 1.
    Aceto, L., Bouyer, P., Burgueño, A., Larsen, K.G.: The power of reachability testing for timed automata. In: Arvind, V., Ramanujam, S. (eds.) FSTTCS 1998. LNCS, vol. 1530, pp. 245–256. Springer, Heidelberg (1998). Scholar
  2. 2.
    Alur, R., Courcoubetis, C., Dill, D.L.: Model-checking in dense real-time. Inf. Comput. 104(1), 2–34 (1993). Scholar
  3. 3.
    Alur, R., Dill, D.L.: A theory of timed automata. Theoret. Comput. Sci. 126(2), 183–235 (1994). Scholar
  4. 4.
    Alur, R., Henzinger, T.A., Vardi, M.Y.: Parametric real-time reasoning. In: Kosaraju, S.R., Johnson, D.S., Aggarwal, A. (eds.) STOC, pp. 592–601. ACM, New York (1993).
  5. 5.
    André, É.: Parametric deadlock-freeness checking timed automata. In: Sampaio, A., Wang, F. (eds.) ICTAC 2016. LNCS, vol. 9965, pp. 469–478. Springer, Cham (2016). Scholar
  6. 6.
    André, É.: A benchmark library for parametric timed model checking. In: Artho, C., Ölveczky, P.C. (eds.) FTSCS 2018. CCIS, vol. 1008, pp. 75–83. Springer, Cham (2019). Scholar
  7. 7.
    André, É.: What’s decidable about parametric timed automata? Int. J. Softw. Tools Technol. Transf. 21(2), 203–219 (2019). Scholar
  8. 8.
    André, É., Bloemen, V., Petrucci, L., van de Pol, J.: Minimal-time synthesis for parametric timed automata. In: Vojnar, T., Zhang, L. (eds.) TACAS 2019. LNCS, vol. 11428, pp. 211–228. Springer, Cham (2019). Scholar
  9. 9.
    André, É., Chatain, T., Encrenaz, E., Fribourg, L.: An inverse method for parametric timed automata. Int. J. Found. Comput. Sci. 20(5), 819–836 (2009). Scholar
  10. 10.
    André, É., Fribourg, L., Kühne, U., Soulat, R.: IMITATOR 2.5: a tool for analyzing robustness in scheduling problems. In: Giannakopoulou, D., Méry, D. (eds.) FM 2012. LNCS, vol. 7436, pp. 33–36. Springer, Heidelberg (2012). Scholar
  11. 11.
    André, É., Hasuo, I., Waga, M.: Offline timed pattern matching under uncertainty. In: Lin, A.W., Sun, J. (eds.) ICECCS, pp. 10–20. IEEE CPS (2018).
  12. 12.
    André, É., Lime, D.: Liveness in L/U-parametric timed automata. In: Legay, A., Schneider, K. (eds.) ACSD, pp. 9–18. IEEE (2017).
  13. 13.
    André, É., Lime, D., Ramparison, M.: TCTL model checking lower/upper-bound parametric timed automata without invariants. In: Jansen, D.N., Prabhakar, P. (eds.) FORMATS 2018. LNCS, vol. 11022, pp. 37–52. Springer, Cham (2018). Scholar
  14. 14.
    André, É., Lime, D., Roux, O.H.: Decision problems for parametric timed automata. In: Ogata, K., Lawford, M., Liu, S. (eds.) ICFEM 2016. LNCS, vol. 10009, pp. 400–416. Springer, Cham (2016). Scholar
  15. 15.
    André, É., Lipari, G., Nguyen, H.G., Sun, Y.: Reachability preservation based parameter synthesis for timed automata. In: Havelund, K., Holzmann, G., Joshi, R. (eds.) NFM 2015. LNCS, vol. 9058, pp. 50–65. Springer, Cham (2015). Scholar
  16. 16.
    André, É., Liu, Y., Sun, J., Dong, J.S., Lin, S.-W.: PSyHCoS: parameter synthesis for hierarchical concurrent real-time systems. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 984–989. Springer, Heidelberg (2013). Scholar
  17. 17.
    André, É., Markey, N.: Language preservation problems in parametric timed automata. In: Sankaranarayanan, S., Vicario, E. (eds.) FORMATS 2015. LNCS, vol. 9268, pp. 27–43. Springer, Cham (2015). Scholar
  18. 18.
    André, É., Nguyen, H.G., Petrucci, L., Sun, J.: Parametric model checking timed automata under non-zenoness assumption. In: Barrett, C., Davies, M., Kahsai, T. (eds.) NFM 2017. LNCS, vol. 10227, pp. 35–51. Springer, Cham (2017). Scholar
  19. 19.
    André, É., Soulat, R.: The Inverse Method. FOCUS Series in Computer Engineering and Information Technology, ISTE Ltd and Wiley, 176 p. (2013)Google Scholar
  20. 20.
    Andreychenko, A., Magnin, M., Inoue, K.: Analyzing resilience properties in oscillatory biological systems using parametric model checking. Biosystems 149, 50–58 (2016). Selected Papers from the Computational Methods in Systems Biology 2015 ConferenceCrossRefGoogle Scholar
  21. 21.
    Aştefănoaei, L., Bensalem, S., Bozga, M., Cheng, C.-H., Ruess, H.: Compositional parameter synthesis. In: Fitzgerald, J., Heitmeyer, C., Gnesi, S., Philippou, A. (eds.) FM 2016. LNCS, vol. 9995, pp. 60–68. Springer, Cham (2016). Scholar
  22. 22.
    Bagnara, R., Hill, P.M., Zaffanella, E.: The parma polyhedra library: toward a complete set of numerical abstractions for the analysis and verification of hardware and software systems. Sci. Comput. Program. 72(1–2), 3–21 (2008). Scholar
  23. 23.
    Beneš, N., Bezděk, P., Larsen, K.G., Srba, J.: Language emptiness of continuous-time parametric timed automata. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9135, pp. 69–81. Springer, Heidelberg (2015). Scholar
  24. 24.
    Bérard, B., Cassez, F., Haddad, S., Lime, D., Roux, O.H.: Comparison of the expressiveness of timed automata and time petri nets. In: Pettersson, P., Yi, W. (eds.) FORMATS 2005. LNCS, vol. 3829, pp. 211–225. Springer, Heidelberg (2005). Scholar
  25. 25.
    Bouyer, P., Markey, N., Sankur, O.: Robustness in timed automata. In: Abdulla, P.A., Potapov, I. (eds.) RP 2013. LNCS, vol. 8169, pp. 1–18. Springer, Heidelberg (2013). Scholar
  26. 26.
    Bozzelli, L., La Torre, S.: Decision problems for lower/upper bound parametric timed automata. Formal Methods Syst. Design 35(2), 121–151 (2009). Scholar
  27. 27.
    Bryant, R.E.: Graph-based algorithms for Boolean function manipulation. IEEE Trans. Comput. 35(8), 677–691 (1986). Scholar
  28. 28.
    Bundala, D., Ouaknine, J.: Advances in parametric real-time reasoning. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds.) MFCS 2014. LNCS, vol. 8634, pp. 123–134. Springer, Heidelberg (2014). Scholar
  29. 29.
    Burch, J.R., Clarke, E.M., McMillan, K.L., Dill, D.L., Hwang, L.J.: Symbolic model checking: 10\(^{20}\) states and beyond. In: LICS, pp. 428–439. IEEE Computer Society (1990).
  30. 30.
    Chevallier, R., Encrenaz-Tiphène, E., Fribourg, L., Xu, W.: Timed verification of the generic architecture of a memory circuit using parametric timed automata. Formal Methods Syst. Des. 34(1), 59–81 (2009). Scholar
  31. 31.
    David, N.: Discrete parameters in Petri nets. Ph.D. thesis. University of Nantes, France (2017)Google Scholar
  32. 32.
    David, N., Jard, C., Lime, D., Roux, O.H.: Discrete parameters in Petri nets. In: Devillers, R., Valmari, A. (eds.) PETRI NETS 2015. LNCS, vol. 9115, pp. 137–156. Springer, Cham (2015). Scholar
  33. 33.
    David, N., Jard, C., Lime, D., Roux, O.H.: Coverability synthesis in parametric Petri nets. In: Meyer, R., Nestmann, U. (eds.) CONCUR. LIPIcs, Dagstuhl Publishing (2017).
  34. 34.
    Delahaye, B.: Consistency for parametric interval Markov chains. In: André, É., Frehse, G. (eds.) SynCoP. OASICS, vol. 44, pp. 17–32. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015).
  35. 35.
    Delahaye, B., Larsen, K.G., Legay, A., Pedersen, M.L., Wasowski, A.: Consistency and refinement for interval Markov chains. J. Log. Algebr. Program. 81(3), 209–226 (2012). Scholar
  36. 36.
    Delahaye, B., Lime, D., Petrucci, L.: Parameter synthesis for parametric interval Markov chains. In: Jobstmann, B., Leino, K.R.M. (eds.) VMCAI 2016. LNCS, vol. 9583, pp. 372–390. Springer, Heidelberg (2016). Scholar
  37. 37.
    Demri, S.: On selective unboundedness of VASS. J. Comput. Syst. Sci. 79(5), 689–713 (2013). Scholar
  38. 38.
    Di Giampaolo, B., La Torre, S., Napoli, M.: Parametric metric interval temporal logic. Theoret. Comput. Sci. 564, 131–148 (2015). Scholar
  39. 39.
    Doyen, L.: Robust parametric reachability for timed automata. Inf. Process. Lett. 102(5), 208–213 (2007). Scholar
  40. 40.
    Fanchon, L., Jacquemard, F.: Formal timing analysis of mixed music scores. In: ICMC. Michigan Publishing, August 2013Google Scholar
  41. 41.
    Frehse, G.: PHAVer: algorithmic verification of hybrid systems past HyTech. Int. J. Softw. Tools Technol. Transf. 10(3), 263–279 (2008). Scholar
  42. 42.
    Frehse, G., et al.: SpaceEx: scalable verification of hybrid systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 379–395. Springer, Heidelberg (2011). Scholar
  43. 43.
    Fribourg, L., Lesens, D., Moro, P., Soulat, R.: Robustness analysis for scheduling problems using the inverse method. In: Reynolds, M., Terenziani, P., Moszkowski, B. (eds.) TIME, pp. 73–80. IEEE Computer Society Press, September 2012.
  44. 44.
    Geeraerts, G., Heußner, A., Praveen, M., Raskin, J.: \(\omega \)-Petri nets: algorithms and complexity. Fundamenta Informaticae 137(1), 29–60 (2015)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Glabbeek, R.J.: The linear time - branching time spectrum. In: Baeten, J.C.M., Klop, J.W. (eds.) CONCUR 1990. LNCS, vol. 458, pp. 278–297. Springer, Heidelberg (1990). Scholar
  46. 46.
    Henzinger, T.A., Ho, P.H., Wong-Toi, H.: HyTech: a model checker for hybrid systems. Int. J. Softw. Tools Technol. Transf. 1(1–2), 110–122 (1997). Scholar
  47. 47.
    Hoare, C.: Communicating Sequential Processes. International Series in Computer Science. Prentice-Hall, Upper Saddle River (1985)zbMATHGoogle Scholar
  48. 48.
    Hune, T., Romijn, J., Stoelinga, M., Vaandrager, F.W.: Linear parametric model checking of timed automata. J. Log. Algebr. Program. 52–53, 183–220 (2002). Scholar
  49. 49.
    Huth, M., Ryan, M.: Logic in Computer Science: Modelling and Reasoning about Systems. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  50. 50.
    Jovanović, A., Lime, D., Roux, O.H.: Integer parameter synthesis for real-time systems. IEEE Trans. Softw. Eng. 41(5), 445–461 (2015). Scholar
  51. 51.
    Karp, R.M., Miller, R.E.: Parallel program schemata. J. Comput. Syst. Sci. 3(2), 147–195 (1969). Scholar
  52. 52.
  53. 53.
    Knapik, M., Meski, A., Penczek, W.: Action synthesis for branching time logic: theory and applications. ACM Trans. Embed. Comput. 14(4), 64 (2015). Scholar
  54. 54.
    Knapik, M., Penczek, W.: Bounded model checking for parametric timed automata. Trans. Petri Nets Other Models Concurr. 5, 141–159 (2012). Scholar
  55. 55.
    Knapik, M., Penczek, W.: Fixed-point methods in parametric model checking. In: Angelov, P., et al. (eds.) Intelligent Systems’2014. AISC, vol. 322, pp. 231–242. Springer, Cham (2015). Scholar
  56. 56.
    Li, J., Sun, J., Gao, B., André, É.: Classification-based parameter synthesis for parametric timed automata. In: Duan, Z., Ong, L. (eds.) ICFEM 2017. LNCS, vol. 10610, pp. 243–261. Springer, Cham (2017). Scholar
  57. 57.
    Lime, D., Roux, O.H., Seidner, C., Traonouez, L.-M.: Romeo: a parametric model-checker for Petri nets with stopwatches. In: Kowalewski, S., Philippou, A. (eds.) TACAS 2009. LNCS, vol. 5505, pp. 54–57. Springer, Heidelberg (2009). Scholar
  58. 58.
    Luthmann, L., Stephan, A., Bürdek, J., Lochau, M.: Modeling and testing product lines with unbounded parametric real-time constraints. In: Cohen, M.B., et al. (eds.) SPLC, vol. A, pp. 104–113. ACM (2017).
  59. 59.
    Miller, J.S.: Decidability and complexity results for timed automata and semi-linear hybrid automata. In: Lynch, N., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 296–310. Springer, Heidelberg (2000). Scholar
  60. 60.
    Minsky, M.L.: Computation: Finite and Infinite Machines. Prentice-Hall Inc., Upper Saddle River (1967)zbMATHGoogle Scholar
  61. 61.
    Parquier, B., et al.: Applying parametric model-checking techniques for reusing real-time critical systems. In: Artho, C., Ölveczky, P.C. (eds.) FTSCS 2016. CCIS, vol. 694, pp. 129–144. Springer, Cham (2017). Scholar
  62. 62.
    Pecheur, C., Raimondi, F.: Symbolic model checking of logics with actions. In: Edelkamp, S., Lomuscio, A. (eds.) MoChArt 2006. LNCS (LNAI), vol. 4428, pp. 113–128. Springer, Heidelberg (2007). Scholar
  63. 63.
    Petrucci, L., van de Pol, J.: Parameter synthesis algorithms for parametric interval Markov chains. In: Baier, C., Caires, L. (eds.) FORTE 2018. LNCS, vol. 10854, pp. 121–140. Springer, Cham (2018). Scholar
  64. 64.
    Raimondi, F., Lomuscio, A.: Automatic verification of multi-agent systems by model checking via ordered binary decision diagrams. J. Appl. Log. 5(2), 235–251 (2007). Scholar
  65. 65.
    Sankur, O.: Symbolic quantitative robustness analysis of timed automata. In: Baier, C., Tinelli, C. (eds.) TACAS 2015. LNCS, vol. 9035, pp. 484–498. Springer, Heidelberg (2015). Scholar
  66. 66.
    Seidner, C.: Vérification des EFFBDs: model checking en Ingénierie Système. (EFFBDs verification: model checking in systems engineering). Ph.D. thesis. University of Nantes, France (2009).
  67. 67.
    Somenzi, F.: CUDD: CU decision diagram package - release 2.5.0.
  68. 68.
    Sun, J., Liu, Y., Dong, J.S., Liu, Y., Shi, L., André, É.: Modeling and verifying hierarchical real-time systems using stateful timed CSP. ACM Trans. Softw. Eng. Methodol. 22(1), 3:1–3:29 (2013). Scholar
  69. 69.
    Sun, Y., André, É., Lipari, G.: Verification of two real-time systems using parametric timed automata. In: Quinton, S., Vardanega, T. (eds.) WATERS, July 2015Google Scholar
  70. 70.
    Traonouez, L.M., Lime, D., Roux, O.H.: Parametric model-checking of stopwatch Petri nets. J. Univ. Comput. Sci. 15(17), 3273–3304 (2009). Scholar
  71. 71.
    Valk, R., Jantzen, M.: The residue of vector sets with applications to decidability problems in Petri nets. Acta Informatica 21(6), 643–674 (1985). Scholar

Copyright information

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

Authors and Affiliations

  1. 1.LIPN, CNRS UMR 7030Université Paris 13VilletaneuseFrance
  2. 2.Institute of Computer Science, PASWarsawPoland
  3. 3.École Centrale de Nantes, LS2N, CNRS UMR 6004NantesFrance
  4. 4.University of Natural Sciences and Humanities, IISiedlcePoland

Personalised recommendations