Skip to main content

Improving SMT Solver Integrations for the Validation of B and Event-B Models

  • 338 Accesses

Part of the Lecture Notes in Computer Science book series (LNPSE,volume 12863)

Abstract

ProB provides a constraint solver for the B-method written in Prolog and optionally can make use of different backends based on SAT or SMT solving. One such solver integration translates B and Event-B operators to SMT-LIB using the C interface of the Z3 solver. This translation uses quantifiers to axiomatise operators when translating to SMT-LIB, which are not well-handled by Z3. Several relational constraints such as the transitive closure are not supported since their translations were too involved.

In this paper, we substantially improve the translation to SMT-LIB by employing a more constructive rather than axiomatised style using Z3’s lambda functions. Thereby, we are able to translate more set-theoretic B and Event-B operators to SMT-LIB, and improve the overall performance. We further extend ProB’s interface to Z3 to run different solver configurations in parallel, e.g., either using the former or new translation. Empirical results show that the new translation to SMT-LIB and the parallel integration of different configurations of the Z3 solver have improved the performance of constraint solving.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-85248-1_7
  • Chapter length: 19 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   64.99
Price excludes VAT (USA)
  • ISBN: 978-3-030-85248-1
  • 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   84.99
Price excludes VAT (USA)
Fig. 1.

Notes

  1. 1.

    Z3 throws the error “internalization of exists is not supported”.

  2. 2.

    The benchmarks can be found in the following repository to reproduce the results: https://github.com/Joshua27/fmics2021_benchmarks.

References

  1. Abbassi, A., Day, N.A., Rayside, D.: Astra version 1.0: evaluating translations from alloy to SMT-LIB. CoRR, abs/1906.05881 (2019)

    Google Scholar 

  2. Abrial, J.-R.: The B-Book: Assigning Programs to Meanings. Cambridge University Press, New York (1996)

    Google Scholar 

  3. Abrial, J.-R.: Modeling in Event-B: System and Software Engineering, 1st edn. Cambridge University Press, New York (2010)

    Google Scholar 

  4. Abrial, J.-R., Mussat, L.: On using conditional definitions in formal theories. In: Bert, D., Bowen, J.P., Henson, M.C., Robinson, K. (eds.) ZB 2002. LNCS, vol. 2272, pp. 242–269. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-45648-1_13

  5. Barrett, C., Fontaine, P., Tinelli, C.: The Satisfiability Modulo Theories Library (SMT-LIB) (2016). www.SMT-LIB.org

  6. Barrett, C.W., Sebastiani, R., Seshia, S.A., Tinelli, C.: Satisfiability modulo theories. In: Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185, pp. 825–885. IOS Press (2009)

    Google Scholar 

  7. Boniol, F., Wiels, V.: The landing gear system case study. In: Boniol, F., Wiels, V., Ait Ameur, Y., Schewe, K.-D. (eds.) ABZ 2014. CCIS, vol. 433, pp. 1–18. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-07512-9_1

  8. Boute, R.: The Euclidean definition of the functions div and mod. ACM Trans. Program. Lang. Syst. 14, 127–144 (1992)

    Google Scholar 

  9. Bride, H., Kouchnarenko, O., Peureux, F., Voiron, G.: Workflow nets verification: SMT or CLP? In: ter Beek, M.H., Gnesi, S., Knapp, A. (eds.) AVoCS 2016, FMICS 2016. LNCS, vol. 9933, pp. 39–55. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-45943-1_3

  10. Carlsson, M., Mildner, P.: SICStus prolog-the first 25 years. Theory Pract. Log. Program. 12(1–2), 35–66 (2012)

    Google Scholar 

  11. Carlsson, M., Ottosson, G., Carlson, B.: An open-ended finite domain constraint solver. In: Glaser, H., Hartel, P., Kuchen, H. (eds.) PLILP 1997. LNCS, vol. 1292, pp. 191–206. Springer, Heidelberg (1997). https://doi.org/10.1007/BFb0033845

  12. de Moura, L., Bjørner, N.: Generalized, efficient array decision procedures. In: 2009 Formal Methods in Computer-Aided Design, pp. 45–52 (2009)

    Google Scholar 

  13. de Moura, L.M., Bjørner, N., Z3: an efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78800-3_2

  14. Déharbe, D., Fontaine, P., Guyot, Y., Voisin, L.: SMT solvers for Rodin. In: Derrick, et al. (eds.) ABZ 2012. LNCS, vol. 7316, pp. 194–207. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-30885-7_14

  15. Déharbe, D.: Integration of SMT-solvers in B and Event-B development environments. Sci. Comput. Program. 78(3), 310–326 (2013). Abstract State Machines, Alloy, B and Z - Selected Papers from ABZ 2010

    Google Scholar 

  16. El Ghazi, A.A., Taghdiri, M.: Relational reasoning via SMT solving. In: Butler, M., Schulte, W. (eds.) FM 2011. LNCS, vol. 6664, pp. 133–148. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-21437-0_12

  17. Hansen, D., Ladenberger, L., Wiegard, H., Bendisposto, J., Leuschel, M.: Validation of the ABZ landing gear system using ProB. In: Boniol, F., Wiels, V., Ait Ameur, Y., Schewe, K.-D. (eds.) ABZ 2014. LNCS, vol. 433, pp. 66–79. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-07512-9_5

  18. Hansen, D., Leuschel, M.: Translating TLA\(^{+}\) to B for validation with ProB. In: Derrick, J., Gnesi, S., Latella, D., Treharne, H. (eds.) IFM 2012. LNCS, vol. 7321, pp. 24–38. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-30729-4_3

  19. Hansen, D., Leuschel, M.: Translating B to TLA\(^{+}\) for validation with TLC. In: Ait Ameur, Y., Schewe, K.D. (eds.) ABZ 2014. LNCS, vol. 8477, pp. 40–55. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-43652-3_4

  20. Hoang, T.S., Snook, C., Ladenberger, L., Butler, M.: Validating the requirements and design of a hemodialysis machine using iUML-B, BMotion Studio, and co-simulation. In: Butler, M., Schewe, K.-D., Mashkoor, A., Biro, M. (eds.) ABZ 2016. LNCS, vol. 9675, pp. 360–375. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-33600-8_31

  21. Howe, J.M., King, A.: A pearl on SAT solving in Prolog. In: Blume, M., Kobayashi, N., Vidal, G. (eds.) FLOPS 2010. LNCS, vol. 6009, pp. 165–174. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12251-4_13

  22. Konnov, I., Kukovec, J., Tran, T.-H.: TLA\(^{+}\) model checking made symbolic. Proc. ACM Program. Lang. 3(OOPSLA), 1–30 (2019)

    Google Scholar 

  23. Krings, S.: Towards infinite-state symbolic model checking for B and event-B. Ph.D. thesis, University of Düsseldorf, Germany (2017)

    Google Scholar 

  24. Krings, S., Leuschel, M.: SMT solvers for validation of B and event-B models. In: Ábrahám, E., Huisman, M. (eds.) IFM 2016. LNCS, vol. 9681, pp. 361–375. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-33693-0_23

  25. Krings, S., Leuschel, M.: Proof assisted bounded and unbounded symbolic model checking of software and system models. Sci. Comput. Prog. 158, 41–63 (2018)

    Google Scholar 

  26. Lamport, L.: Specifying Systems: The TLA\(^{+}\) Language and Tools for Hardware and Software Engineers. Addison-Wesley Longman Publishing Co., Inc, Boston (2002)

    Google Scholar 

  27. Leuschel, M., Butler, M.: ProB: a model checker for B. In: Araki, K., Gnesi, S., Mandrioli, D. (eds.) FME 2003. LNCS, vol. 2805, pp. 855–874. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-45236-2_46

  28. Leuschel, M., Butler, M.: ProB: an automated analysis toolset for the B method. Int. J. Softw. Tools Technol. Transf. 10(2), 185–203 (2008)

    Google Scholar 

  29. Mann, M., Wilson, A., Tinelli, C., Barrett, C.W.: SMT-switch: a solver-agnostic C++ API for SMT solving. CoRR, abs/2007.01374 (2020)

    Google Scholar 

  30. Mashkoor, A.: The hemodialysis machine case study. In: Butler, M., Schewe, K.-D., Mashkoor, A., Biro, M. (eds.) ABZ 2016. LNCS, vol. 9675, pp. 329–343. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-33600-8_29

  31. Nieuwenhuis, R., Oliveras, A., Tinelli, C.: Abstract DPLL and abstract DPLL modulo theories. In: Baader, F., Voronkov, A. (eds.) LPAR 2005. LNCS, vol. 6452, pp. 36–50. Springer, Heidelberg (2005). https://doi.org/10.1007/978-3-540-32275-7_3

  32. Nieuwenhuis, R., Oliveras, A., Tinelli, C.: Solving SAT and SAT modulo theories: from an abstract Davis-Putnam-Logemann-Loveland procedure to DPLL(T). J. ACM 53(6), 937–977 (2006)

    Google Scholar 

  33. Plagge, D., Leuschel, M.: Validating B, Z and TLA\(^{+}\) using ProB and Kodkod. In: FM 2012. LNCS, vol. 7436, pp. 372–386. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32759-9_31

  34. Silva, J.P.M., Lynce, I., Malik, S.: Conflict-driven clause learning SAT solvers. In: Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185, pp. 131–153. IOS Press (2009)

    Google Scholar 

  35. Silva, J.P.M., Sakallah, K.A.: GRASP - a new search algorithm for satisfiability. In: Proceedings of the 1996 IEEE/ACM International Conference on Computer-Aided Design, ICCAD 1996, USA, pp. 220–227. IEEE Computer Society (1997)

    Google Scholar 

  36. Torlak, E., Jackson, D.: Kodkod: a relational model finder. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 632–647. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-71209-1_49

  37. Weber, T.: SMT solvers: new oracles for the HOL theorem prover. Int. J. Softw. Tools Technol. Transf. (STTT) 13(5), 419–429 (2011)

    Google Scholar 

  38. Weber, T., Conchon, S., Déharbe, D., Heizmann, M., Niemetz, A., Reger, G.: The SMT competition 2015–2018. J. Satisf. Boolean Model. Comput. 11(1), 221–259 (2019)

    Google Scholar 

  39. Yu, Y., Manolios, P., Lamport, L.: Model checking TLA\(^{+}\) specifications. In: Pierre, L., Kropf, T. (eds.) CHARME 1999. LNCS, vol. 1703, pp. 54–66. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48153-2_6

Download references

Acknowledgements

We would like to thank the reviewers of FMICS’2021 for their useful feedback.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Joshua Schmidt .

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

Schmidt, J., Leuschel, M. (2021). Improving SMT Solver Integrations for the Validation of B and Event-B Models. In: Lluch Lafuente, A., Mavridou, A. (eds) Formal Methods for Industrial Critical Systems. FMICS 2021. Lecture Notes in Computer Science(), vol 12863. Springer, Cham. https://doi.org/10.1007/978-3-030-85248-1_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-85248-1_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-85247-4

  • Online ISBN: 978-3-030-85248-1

  • eBook Packages: Computer ScienceComputer Science (R0)