Advertisement

Invariant Checking of NRA Transition Systems via Incremental Reduction to LRA with EUF

  • Alessandro Cimatti
  • Alberto Griggio
  • Ahmed Irfan
  • Marco Roveri
  • Roberto Sebastiani
Conference paper
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10205)

Abstract

Model checking invariant properties of designs, represented as transition systems, with non-linear real arithmetic (NRA), is an important though very hard problem. On the one hand NRA is a hard-to-solve theory; on the other hand most of the powerful model checking techniques lack support for NRA. In this paper, we present a counterexample-guided abstraction refinement (CEGAR) approach that leverages linearization techniques from differential calculus to enable the use of mature and efficient model checking algorithms for transition systems on linear real arithmetic (LRA) with uninterpreted functions (EUF). The results of an empirical evaluation confirm the validity and potential of this approach.

This is a preview of subscription content, log in to check access.

Notes

Acknowledgement

We greatly thank the iSAT3 team for providing the latest iSAT3 executable and iSAT3-CFG benchmarks. We also thank James Davenport for the fruitful discussions on CAD techniques and finding solutions in NRA.

References

  1. 1.
    Ábrahám, E., Corzilius, F., Loup, U., Sturm, T.: A lazy SMT-solver for a non-linear subset of real algebra. In: Dagstuhl Seminar Proceedings. Schloss Dagstuhl-Leibniz-Zentrum f A1/4r Informatik (2010)Google Scholar
  2. 2.
    Bak, S., Bogomolov, S., Johnson, T.T.: HYST: a source transformation and translation tool for hybrid automaton models. In: Proceedings of the 18th International Conference on Hybrid Systems: Computation and Control, pp. 128–133. ACM (2015)Google Scholar
  3. 3.
    Biere, A., Cimatti, A., Clarke, E., Zhu, Y.: Symbolic model checking without BDDs. In: Cleaveland, W.R. (ed.) TACAS 1999. LNCS, vol. 1579, pp. 193–207. Springer, Heidelberg (1999). doi: 10.1007/3-540-49059-0_14 CrossRefGoogle Scholar
  4. 4.
    Birgmeier, J., Bradley, A.R., Weissenbacher, G.: Counterexample to induction-guided abstraction-refinement (CTIGAR). In: Biere, A., Bloem, R. (eds.) CAV 2014. LNCS, vol. 8559, pp. 831–848. Springer, Cham (2014). doi: 10.1007/978-3-319-08867-9_55 Google Scholar
  5. 5.
    Brain, M., D’Silva, V., Griggio, A., Haller, L., Kroening, D.: Interpolation-based verification of floating-point programs with abstract CDCL. In: Logozzo, F., Fähndrich, M. (eds.) SAS 2013. LNCS, vol. 7935, pp. 412–432. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-38856-9_22 CrossRefGoogle Scholar
  6. 6.
    Brat, G., Bushnell, D., Davies, M., Giannakopoulou, D., Howar, F., Kahsai, T.: Verifying the safety of a flight-critical system. In: Bjørner, N., de Boer, F. (eds.) FM 2015. LNCS, vol. 9109, pp. 308–324. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-19249-9_20 CrossRefGoogle Scholar
  7. 7.
    Cavada, R., et al.: The nuXmv symbolic model checker. In: Biere, A., Bloem, R. (eds.) CAV 2014. LNCS, vol. 8559, pp. 334–342. Springer, Cham (2014). doi: 10.1007/978-3-319-08867-9_22 Google Scholar
  8. 8.
    Champion, A., Gurfinkel, A., Kahsai, T., Tinelli, C.: CoCoSpec: a mode-aware contract language for reactive systems. In: De Nicola, R., Kühn, E. (eds.) SEFM 2016. LNCS, vol. 9763, pp. 347–366. Springer, Cham (2016). doi: 10.1007/978-3-319-41591-8_24 Google Scholar
  9. 9.
    Cimatti, A., Griggio, A., Mover, S., Tonetta, S.: IC3 modulo theories via implicit predicate abstraction. In: Ábrahám, E., Havelund, K. (eds.) TACAS 2014. LNCS, vol. 8413, pp. 46–61. Springer, Heidelberg (2014). doi: 10.1007/978-3-642-54862-8_4 CrossRefGoogle Scholar
  10. 10.
    Cimatti, A., Griggio, A., Mover, S., Tonetta, S.: HyComp: an SMT-based model checker for hybrid systems. In: Baier, C., Tinelli, C. (eds.) TACAS 2015. LNCS, vol. 9035, pp. 52–67. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-46681-0_4 Google Scholar
  11. 11.
    Cimatti, A., Mover, S., Tonetta, S.: A quantifier-free SMT encoding of non-linear hybrid automata. In: Formal Methods in Computer-Aided Design (FMCAD), pp. 187–195. IEEE (2012)Google Scholar
  12. 12.
    Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition-preliminary report. SIGSAM Bull. 8(3), 80–90 (1974). http://doi.acm.org/10.1145/1086837.1086852 CrossRefGoogle Scholar
  13. 13.
    Moura, L., 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). doi: 10.1007/978-3-540-78800-3_24 CrossRefGoogle Scholar
  14. 14.
    Dutertre, B.: Yices 2.2. In: Biere, A., Bloem, R. (eds.) CAV 2014. LNCS, vol. 8559, pp. 737–744. Springer, Heidelberg (2014). doi: 10.1007/978-3-319-08867-9_49 Google Scholar
  15. 15.
    Eén, N., Sörensson, N.: Temporal induction by incremental SAT solving. Electron. Notes Theor. Comput. Sci. 89(4), 543–560 (2003)CrossRefMATHGoogle Scholar
  16. 16.
    Gao, S., Kong, S., Clarke, E.M.: dReal: an SMT solver for nonlinear theories over the reals. In: Bonacina, M.P. (ed.) CADE 2013. LNCS (LNAI), vol. 7898, pp. 208–214. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-38574-2_14 CrossRefGoogle Scholar
  17. 17.
    Gario, M., Micheli, A.: PySMT: a solver-agnostic library for fast prototyping of SMT-based algorithms. In: Proceedings of the 13th International Workshop on Satisfiability Modulo Theories (SMT), pp. 373–384 (2015)Google Scholar
  18. 18.
    Hoder, K., Bjørner, N.: Generalized property directed reachability. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 157–171. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-31612-8_13 CrossRefGoogle Scholar
  19. 19.
    Hueschen, R.M.: Development of the Transport Class Model (TCM) aircraft simulation from a sub-scale Generic Transport Model (GTM) simulation. Technical report, NASA Langley Research Center (2011)Google Scholar
  20. 20.
    Jovanović, D., Moura, L.: Solving non-linear arithmetic. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS (LNAI), vol. 7364, pp. 339–354. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-31365-3_27 CrossRefGoogle Scholar
  21. 21.
    Komuravelli, A., Gurfinkel, A., Chaki, S.: SMT-based model checking for recursive programs. Form. Methods Syst. Des. 48(3), 175–205 (2016)CrossRefMATHGoogle Scholar
  22. 22.
    Kong, S., Gao, S., Chen, W., Clarke, E.: dReach: \({\delta }\)-reachability analysis for hybrid systems. In: Baier, C., Tinelli, C. (eds.) TACAS 2015. LNCS, vol. 9035, pp. 200–205. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-46681-0_15 Google Scholar
  23. 23.
    Kupferschmid, S., Becker, B.: Craig interpolation in the presence of non-linear constraints. In: Fahrenberg, U., Tripakis, S. (eds.) FORMATS 2011. LNCS, vol. 6919, pp. 240–255. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-24310-3_17 CrossRefGoogle Scholar
  24. 24.
    Mahdi, A., Scheibler, K., Neubauer, F., Fränzle, M., Becker, B.: Advancing software model checking beyond linear arithmetic theories. In: Bloem, R., Arbel, E. (eds.) HVC 2016. LNCS, vol. 10028, pp. 186–201. Springer, Heidelberg (2016). doi: 10.1007/978-3-319-49052-6_12 CrossRefGoogle Scholar
  25. 25.
    Maréchal, A., Fouilhé, A., King, T., Monniaux, D., Périn, M.: Polyhedral approximation of multivariate polynomials using Handelman’s theorem. In: Jobstmann, B., Leino, K.R.M. (eds.) VMCAI 2016. LNCS, vol. 9583, pp. 166–184. Springer, Heidelberg (2016). doi: 10.1007/978-3-662-49122-5_8 CrossRefGoogle Scholar
  26. 26.
    McMillan, K.L.: Interpolation and SAT-based model checking. In: Hunt, W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 1–13. Springer, Heidelberg (2003). doi: 10.1007/978-3-540-45069-6_1 CrossRefGoogle Scholar
  27. 27.
    Nuzzo, P., Puggelli, A., Seshia, S.A., Sangiovanni-Vincentelli, A.: CalCS: SMT solving for non-linear convex constraints. In: Proceedings of the 2010 Conference on Formal Methods in Computer-Aided Design, pp. 71–80. FMCAD Inc. (2010)Google Scholar
  28. 28.
    Scheibler, K., Kupferschmid, S., Becker, B.: Recent improvements in the SMT solver iSAT. MBMV 13, 231–241 (2013)Google Scholar
  29. 29.
    Sheeran, M., Singh, S., Stålmarck, G.: Checking safety properties using induction and a SAT-solver. In: Hunt, W.A., Johnson, S.D. (eds.) FMCAD 2000. LNCS, vol. 1954, pp. 127–144. Springer, Heidelberg (2000). doi: 10.1007/3-540-40922-X_8 CrossRefGoogle Scholar
  30. 30.
    Tiwari, A.: Time-aware abstractions in HybridSal. In: Kroening, D., Păsăreanu, C.S. (eds.) CAV 2015. LNCS, vol. 9206, pp. 504–510. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-21690-4_34 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Alessandro Cimatti
    • 1
  • Alberto Griggio
    • 1
  • Ahmed Irfan
    • 1
    • 2
  • Marco Roveri
    • 1
  • Roberto Sebastiani
    • 2
  1. 1.Fondazione Bruno KesslerTrentoItaly
  2. 2.University of TrentoTrentoItaly

Personalised recommendations