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Distributed parametric model checking timed automata under non-Zenoness assumption

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Abstract

Model checking timed systems may be negatively impacted by the presence of Zeno runs as counterexamples. Such runs contain an infinite number of discrete actions occurring in a finite time, which is unrealistic; therefore, these runs must be pruned during model checking. Given a model of a timed system and a property, parametric model checking aims at synthesizing timing valuations such that the model satisfies the property. A popular formalism for parametric timed model checking is parametric timed automata (PTAs), an extension of timed automata with unknown constants. We address the presence of Zeno runs in parametric model checking. We first show that the emptiness of the set of parameter valuations for which at least one counterexample run being non-Zeno is undecidable for PTAs. As a second contribution, we adopt a more pragmatic approach, and propose a procedure based on a transformation of PTAs into Clock Upper Bound PTAs to derive all valuations whenever it terminates, and some of them otherwise. We implemented our algorithm in IMITATOR, and we show that our method can be distributed over a cluster to further speedup the computation.

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Data availability

The datasets analyzed during the current study are available in the Zenodo repository, at 10.5281/zenodo.7096134.

Notes

  1. Observe that a parametric guard with two upper bounds on \(x\) can be encoded using two consecutive transitions in 0-time.

  2. Our definition is slightly more liberal than the one in [51].

  3. Note that if a clock has more than a single upper bound in a guard, then the minimum can be encoded as a disjunction of constraints, and our results would still apply with non-convex constraints (that can be implemented using a finite list of convex constraints).

  4. This model assumes that, after the change of a signal in the input of a gate, the output changes after a delay which is modeled using a parametric closed interval.

  5. A purely parametric constraint (e. g. \(p_1 > p_2 \wedge p_3 = 3\)) is generally not allowed by the PTA syntax, but can be simulated using appropriate clocks (e. g. \(p_1> x> p_2 \wedge p_3 = x' = 3\)). Such parametric constraints are allowed in the input syntax of IMITATOR.

  6. Following a well-known result for PTAs, all symbolic states belonging to a same cycle in a parametric zone graph necessarily have the same parameter constraint [46].

References

  1. Alur R, Dill DL (1994) A theory of timed automata. Theor Comput Sci 126(2):183–235. https://doi.org/10.1016/0304-3975(94)90010-8

    Article  MathSciNet  MATH  Google Scholar 

  2. Alur R, Henzinger TA, Vardi MY (1993) Parametric real-time reasoning. In: Kosaraju SR, Johnson DS, Aggarwal A (eds) STOC. ACM, New York, NY, USA, pp 592–601. https://doi.org/10.1145/167088.167242

    Chapter  Google Scholar 

  3. André É (2016) Parametric deadlock-freeness checking timed automata. In: Sampaio ACA, Wang F (eds) ICTAC. Lecture Notes in Computer Science, vol. 9965, Springer. pp 469–478. https://doi.org/10.1007/978-3-319-46750-4_27

  4. André É (2019) What’s decidable about parametric timed automata? Int J Softw Tools Technol Transf 21(2):203–219. https://doi.org/10.1007/s10009-017-0467-0

    Article  Google Scholar 

  5. André É, Arcaini P, Gargantini A, Radavelli M (2019) Repairing timed automata clock guards through abstraction and testing. In: Beyer D, Keller C (eds) TAP. Lecture Notes in Computer Science, vol. 11823, Springer. pp 129–146. https://doi.org/10.1007/978-3-030-31157-5_9

  6. André É, Arias J, Petrucci L, van de Pol J (2021) Iterative bounded synthesis for efficient cycle detection in parametric timed automata. In: Groote JF, Larsen KG (eds) TACAS. Lecture Notes in Computer Science, vol. 12651, Springer. pp 311–329. https://doi.org/10.1007/978-3-030-72016-2_17

  7. André É, Chatain Th, De Smet O, Fribourg L, Ruel S (Nov 2009) Synthèse de contraintes temporisées pour une architecture d’automatisation en réseau. In: Lime D, Roux OH (eds) MSR. Journal Européen des Systèmes Automatisés, vol. 43, Hermès. pp 1049–1064

  8. André É, Chatain Th, Encrenaz E, Fribourg L (2009) An inverse method for parametric timed automata. Int J Found. Comput. Sci. 20(5):819–836. https://doi.org/10.1142/S0129054109006905

    Article  MathSciNet  MATH  Google Scholar 

  9. André É, Coti C, Evangelista S (2014) Distributed behavioral cartography of timed automata. In: Dongarra J, Ishikawa Y, Atsushi H (eds) EuroMPI/ASIA. ACM, New York. pp 109–114. https://doi.org/10.1145/2642769.2642784

    Chapter  Google Scholar 

  10. André É, Coti C, Nguyen HG (Nov 2015) Enhanced distributed behavioral cartography of parametric timed automata. In: Butler M, Conchon S, Zaïdi F (eds) ICFEM. Lecture Notes in Computer Science, vol. 9407, Springer. pp 319–335. https://doi.org/10.1007/978-3-319-25423-4_21

  11. André É, Fribourg L, Kühne U, Soulat R (Aug 2012) IMITATOR 2.5: A tool for analyzing robustness in scheduling problems. In: Giannakopoulou D, Méry D (eds) FM. Lecture Notes in Computer Science, vol. 7436, Springer. pp 33–36. https://doi.org/10.1007/978-3-642-32759-9_6

  12. André É, Lime D (2017) Liveness in L/U-parametric timed automata. In: Legay A, Schneider K (eds) ACSD. IEEE. pp 9–18. https://doi.org/10.1109/ACSD.2017.19

  13. André É, Lime D, Markey N (Jan 2020) Language preservation problems in parametric timed automata. Logical Methods in Comput Sci 16(1): https://doi.org/10.23638/LMCS-16(1:5)2020

  14. André É, Lime D, Roux OH (2016) Decision problems for parametric timed automata. In: Ogata K, Lawford M, Liu S (eds) ICFEM. Lecture Notes in Computer Science, vol. 10009, Springer. pp 400–416. https://doi.org/10.1007/978-3-319-47846-3_25

  15. André É, Lin SW (2017) Learning-based compositional parameter synthesis for event-recording automata. In: Bouajjani A, Alexandra S (eds) FORTE. Lecture Notes in Computer Science, vol. 10321, Springer. pp 17–32. https://doi.org/10.1007/978-3-319-60225-7_2

  16. André É, Liu Y, Sun J, Dong JS (2014) Parameter synthesis for hierarchical concurrent real-time systems. Real-Time Syst 50(5–6):620–679. https://doi.org/10.1007/s11241-014-9208-6

    Article  MATH  Google Scholar 

  17. André É, Marinho D, van de Pol J (2021) A benchmarks library for extended timed automata. In: Loulergue F, Wotawa F (eds) TAP. Lecture Notes in Computer Science, vol. 12740, Springer. pp 39–50. https://doi.org/10.1007/978-3-030-79379-1_3

  18. André É, Nguyen HG, Petrucci L, Sun J (2017) Parametric model checking timed automata under non-Zenoness assumption. In: Barrett C, Kahsai T (eds) NFM. Lecture Notes in Computer Science, vol. 10227, Springer. pp 35–51. https://doi.org/10.1007/978-3-319-57288-8_3

  19. Aştefănoaei L, Bensalem S, Bozga M, Cheng C, Ruess H (2016) Compositional parameter synthesis. In: Fitzgerald JS, Heitmeyer CL, Gnesi S, Philippou A (eds) FM. Lecture Notes in Computer Science, vol. 9995, pp 60–68. https://doi.org/10.1007/978-3-319-48989-6_4

  20. Bagnara R, Hill PM, Zaffanella E (2008) 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. https://doi.org/10.1016/j.scico.2007.08.001

    Article  MathSciNet  Google Scholar 

  21. Behrmann G, Bouyer P, Larsen KG, Pelánek R (2006) Lower and upper bounds in zone-based abstractions of timed automata. Int J Softw Tools Technol Transf 8(3):204–215. https://doi.org/10.1007/s10009-005-0190-0

    Article  MATH  Google Scholar 

  22. Beneš N, Bezděk P, Larsen KG, Srba J (Jul 2015) Language emptiness of continuous-time parametric timed automata. In: Halldórsson MM, Iwama K, Kobayashi N, Speckmann B (eds) ICALP, Part II. Lecture Notes in Computer Science, vol. 9135, Springer. pp 69–81. https://doi.org/10.1007/978-3-662-47666-6_6

  23. Bengtsson J, Yi W (2003) Timed automata: Semantics, algorithms and tools. In: Desel J, Reisig W, Rozenberg G (eds) Lectures on Concurrency and Petri Nets, Advances in Petri Nets. Lecture Notes in Computer Science, vol. 3098, Springer. pp 87–124. https://doi.org/10.1007/978-3-540-27755-2_3

  24. Beyer D, Lewerentz C, Noack A (2003) Rabbit: A tool for BDD-based verification of real-time systems. In: Jr, WAH, Somenzi F (eds) CAV. Lecture Notes in Computer Science, vol. 2725, Springer. pp 122–125. https://doi.org/10.1007/978-3-540-45069-6_13

  25. Bowman H, Gómez R (2006) How to stop time stopping. Form Asp Comput 18(4):459–493. https://doi.org/10.1007/s00165-006-0010-7

    Article  MATH  Google Scholar 

  26. Bozga M, Daws C, Maler O, Olivero A, Tripakis S, Yovine S (1998) Kronos: A model-checking tool for real-time systems. In: Hu AJ, Vardi MY (eds) Proceedings of the 10th International Conference on Computer Aided Verification (CAV 1998). Lecture Notes in Computer Science, vol. 1427, Springer. pp 546–550. https://doi.org/10.1007/BFb0028779

  27. Bozzelli L, La Torre S (2009) Decision problems for lower/upper bound parametric timed automata. Form Methods Syst Des 35(2):121–151. https://doi.org/10.1007/s10703-009-0074-0

    Article  MATH  Google Scholar 

  28. Clarisó R, Cortadella J (2005) Verification of concurrent systems with parametric delays using octahedra. In: ACSD. IEEE Computer Society. pp 122–131. https://doi.org/10.1109/ACSD.2005.34

  29. Clarisó R, Cortadella J (2007) The octahedron abstract domain. Sci Comput Program 64(1):115–139. https://doi.org/10.1016/j.scico.2006.03.009

    Article  MathSciNet  MATH  Google Scholar 

  30. Collomb-Annichini A, Sighireanu M (2001) Parameterized reachability analysis of the IEEE 1394 root contention protocol using TReX. In: RT-TOOLS

  31. Dong JS, Hao P, Qin S, Sun J, Yi W (2008) Timed automata patterns. IEEE Trans Softw Eng 34(6):844–859. https://doi.org/10.1109/TSE.2008.52

    Article  Google Scholar 

  32. Evangelista S, Laarman A, Petrucci L, van de Pol J (2012) Improved multi-core nested depth-first search. In: Chakraborty S, Mukund M (eds) ATVA. Lecture Notes in Computer Science, vol. 7561. Springer. pp 269–283. https://doi.org/10.1007/978-3-642-33386-6_22

  33. Gómez R, Bowman H (2007) Efficient detection of Zeno runs in timed automata. In: Raskin JF, Thiagarajan PS (eds) FORMATS. Lecture Notes in Computer Science, vol. 4763, Springer. pp 195–210. https://doi.org/10.1007/978-3-540-75454-1_15

  34. Herbreteau F, Srivathsan B, Walukiewicz I (2012) Efficient emptiness check for timed Büchi automata. Form Methods Syst Des 40(2):122–146. https://doi.org/10.1007/s10703-011-0133-1

    Article  MATH  Google Scholar 

  35. Hune T, Romijn J, Stoelinga M, Vaandrager FW (2002) Linear parametric model checking of timed automata. J Logic Algebraic Program 52–53:183–220. https://doi.org/10.1016/S1567-8326(02)00037-1

    Article  MathSciNet  MATH  Google Scholar 

  36. Jovanović A, Lime D, Roux OH (2015) Integer parameter synthesis for real-time systems. IEEE Trans Softw Eng 41(5):445–461. https://doi.org/10.1109/TSE.2014.2357445

    Article  Google Scholar 

  37. Knapik M, Penczek W (2012) Bounded model checking for parametric timed automata. Trans Petri Nets Other Models Concurr 5:141–159. https://doi.org/10.1007/978-3-642-29072-5_6

    Article  MATH  Google Scholar 

  38. Kwiatkowska MZ, Norman G, Sproston J, Wang F (2007) Symbolic model checking for probabilistic timed automata. Inf Comput 205(7):1027–1077. https://doi.org/10.1016/j.ic.2007.01.004

    Article  MathSciNet  MATH  Google Scholar 

  39. Laarman A, Olesen MC, Dalsgaard AE, Larsen KG, van De Pol J (Jul 2013) Multi-core emptiness checking of timed Büchi automata using inclusion abstraction. In: Sharygina N, Veith H (eds) CAV. Lecture Notes in Computer Science, vol. 8044. Springer. pp 968–983. Heidelberg, Germany. https://doi.org/10.1007/978-3-642-39799-8_69

  40. Larsen KG, Pettersson P, Yi W (1997) UPPAAL in a nutshell. Int J Softw Tools Technol Transf 1(1–2):134–152. https://doi.org/10.1007/s100090050010

    Article  MATH  Google Scholar 

  41. Lipari G, Sun Y, André É, Fribourg L (Apr 2014) Toward parametric timed interfaces for real-time components. In: André E, Frehse G (eds) SynCoP. Electronic Proceedings in Theoretical Computer Science, vol. 145, pp 49–64. https://doi.org/10.4204/EPTCS.145.6, http://rvg.web.cse.unsw.edu.au/eptcs/paper.cgi?145.6.pdf

  42. Luthmann L, Gerecht T, Stephan A, Bürdek J, Lochau M (2019) Minimum/maximum delay testing of product lines with unbounded parametric real-time constraints. J Syst Softw 149:535–553. https://doi.org/10.1016/j.jss.2018.12.028

    Article  Google Scholar 

  43. Luthmann L, Stephan A, Bürdek J, Lochau M (2017) Modeling and testing product lines with unbounded parametric real-time constraints. In: Cohen MB, Acher M, Fuentes L, Schall D, Bosch J, Capilla R, Bagheri E, Xiong Y, Troya J, Cortés AR, Benavides D (eds) SPLC, Volume A, ACM. pp 104–113. https://doi.org/10.1145/3106195.3106204

  44. Miller JS (2000) Decidability and complexity results for timed automata and semi-linear hybrid automata. In: Lynch NA, Krogh BH (eds) HSCC. Lecture Notes in Computer Science, vol. 1790, Springer. pp 296–309. https://doi.org/10.1007/3-540-46430-1_26

  45. Minsky ML (1967) Computation: finite and infinite machines. Prentice-Hall Inc, Upper Saddle River, NJ, USA

    MATH  Google Scholar 

  46. Nguyen HG, Petrucci L, van de Pol J (Dec 2018) Layered and collecting NDFS with subsumption for parametric timed automata. In: Lin AW, Sun J (eds) ICECCS. pp 1–9. IEEE Computer Society. https://doi.org/10.1109/ICECCS2018.2018.00009

  47. Sun J, Liu Y, Dong JS, Pang J (2009) PAT: Towards flexible verification under fairness. In: Bouajjani A, Maler O (eds) CAV. Lecture Notes in Computer Science, vol. 5643, Springer. pp 709–714. https://doi.org/10.1007/978-3-642-02658-4_59

  48. Tripakis S (1999) Verifying progress in timed systems. In: Katoen JP (ed) ARTS. Lecture Notes in Computer Science, vol. 1601. Springer, New York. pp 299–314

  49. Tripakis S, Yovine S, Bouajjani A (2005) Checking timed Büchi automata emptiness efficiently. Form Methods Syst Des 26(3):267–292. https://doi.org/10.1007/s10703-005-1632-8

    Article  MATH  Google Scholar 

  50. Wang F (2001) Symbolic verification of complex real-time systems with clock-restriction diagram. In: Kim M, Chin B, Kang S, Lee D (eds) FORTE. IFIP Conference Proceedings, vol. 197, Kluwer. pp 235–250

  51. Wang T, Sun J, Wang X, Liu Y, Si Y, Dong JS, Yang X, Li X (2015) A systematic study on explicit-state non-Zenoness checking for timed automata. IEEE Trans Softw Eng 41(1):3–18. https://doi.org/10.1109/TSE.2014.2359893

    Article  Google Scholar 

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Acknowledgements

We would like to thank the anonymous reviewers for useful comments.

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Authors and Affiliations

  1. Université de Lorraine, CNRS, Inria, LORIA, Nancy, France

    Étienne André

  2. CEA - LIST, NANO-INNOV, CEA Saclay, 91120, Palaiseau, France

    Hoang Gia Nguyen

  3. Université Sorbonne Paris Nord, LIPN, CNRS UMR 7030, F-93430, Villetaneuse, France

    Laure Petrucci

  4. School of Information Systems, Singapore Management University, Singapore, Singapore

    Jun Sun

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This work is partially supported by the ANR national research program PACS (ANR-14-CE28-0002), by the ANR-NRF French-Singaporean research program ProMiS (ANR-19-CE25-0015) and by the ANR national research program BisoUS.

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André, É., Nguyen, H.G., Petrucci, L. et al. Distributed parametric model checking timed automata under non-Zenoness assumption. Form Methods Syst Des 59, 253–290 (2021). https://doi.org/10.1007/s10703-022-00400-z

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