Software & Systems Modeling

, Volume 14, Issue 1, pp 121–148 | Cite as

Improving the SAT modulo ODE approach to hybrid systems analysis by combining different enclosure methods

  • Andreas Eggers
  • Nacim Ramdani
  • Nedialko S. Nedialkov
  • Martin Fränzle
Special Section Paper

Abstract

Aiming at automatic verification and analysis techniques for hybrid discrete-continuous systems, we present a novel combination of enclosure methods for ordinary differential equations (ODEs) with the iSAT solver for large Boolean combinations of arithmetic constraints. Improving on our previous work, the contribution of this paper lies in combining iSAT with VNODE-LP, as a state-of-the-art interval solver for ODEs, and with bracketing systems, which exploit monotonicity properties allowing to find enclosures for problems that VNODE-LP alone cannot enclose tightly. We apply the combined iSAT-ODE solver to the analysis of a variety of non-linear hybrid systems by solving predicative encodings of reachability properties and of an inductive stability argument, and evaluate the impact of the different enclosure methods, decision heuristics and their combination. Our experiments include classic benchmarks from the literature, as well as a newly-designed conveyor belt system that combines hybrid behavior of parallel components, a slip-stick friction model with non-linear dynamics and flow invariants and several dimensions of parameterization. In the paper, we also present and evaluate an extension of VNODE-LP tailored to its use as a deduction mechanism within iSAT-ODE, to allow fast re-evaluations of enclosures over arbitrary subranges of the analyzed time span.

Keywords

Analysis of hybrid discrete-continuous systems Satisfiability modulo theories Enclosure methods for ODEs Bracketing systems 

Notes

Acknowledgments

We would like to thank Stefan Ratschan, Christian Herde, Tino Teige, Jens Oehlerking, and Corina Mitrohin for discussions on the region-stability-related proof scheme utilized for the experiments in this paper and all colleagues from the transregional research center AVACS, project H1/2 “Constraint-based Verification for Hybrid Systems” for the joint development of the iSAT core. Additionally, we are grateful to the reviewers of [6] for their detailed comments. Especially by insisting on a more thorough experimental evaluation and by pointing out shortcomings in our presentation, the SoSyM reviewers have helped tremendously to improve the quality of this paper. Thank you!

References

  1. 1.
    Berz, M.: COSY INFINITY version 8 reference manual. Tech. Rep. MSUCL-1088, National Superconducting Cyclotron Laboratory, Michigan State University, USA (1997)Google Scholar
  2. 2.
    Clarke, E.M., Fehnker, A., Han, Z., Krogh, B.H., Stursberg, O., Theobald, M.: Verification of hybrid systems based on counterexample-guided abstraction refinement. In: Gravel, H., Hatcliff, J. (eds.) TACAS, Lecture Notes in Computer Science vol 2619, pp. 192–207. Springer, Berlin (2003)Google Scholar
  3. 3.
    Davis, M., Putnam, H.: A computing procedure for quantification theory. J. ACM 7(3), 201–215 (1960)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Davis, M., Logemann, G., Loveland, D.: A machine program for theorem proving. Commun. ACM 5, 394–397 (1962)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Eggers, A., Fränzle, M., Herde, C.: SAT modulo ODE: a direct SAT approach to hybrid systems. In: ATVA, LNCS, vol. 5311, pp. 171–185. Springer, New York (2008)Google Scholar
  6. 6.
    Eggers, A., Ramdani, N., Nedialkov, NS., Fränzle, M.: Improving SAT modulo ODE for hybrid systems analysis by combining different enclosure methods. In: Barthe, G., Pardo, A., Schneider, G. (eds.) Proceedings of the Ninth International Conference on Software Engineering and Formal Methods (SEFM), LNCS, vol. 7041, pp. 172–187. Springer, Berlin (2011). doi: 10.1007/978-3-642-24690-6-13
  7. 7.
    Fousse, L., Hanrot, G., Lefèvre, V., Pélissier, P., Zimmermann, P.: MPFR: a multiple-precision binary floating-point library with correct rounding. ACM Trans. Math. Softw. 33(2) (2007). doi: 10.1145/1236463.1236468, MPFR is available at http://www.mpfr.org/
  8. 8.
    Fränzle, M., Herde, C., Ratschan, S., Schubert, T., Teige, T.: Efficient solving of large non-linear arithmetic constraint systems with complex boolean structure. JSAT 1(3–4), 209–236 (2007)Google Scholar
  9. 9.
    Goldsztejn, A., Mullier, O., Eveillard, D., Hosobe, H.: Including ordinary differential equations based constraints in the standard CP framework. In: Cohen, D. (ed.) Principles and Practice of Constraint Programming—CP 2010, LNCS, vol. 6308, pp. 221–235. Springer, Berlin (2010)CrossRefGoogle Scholar
  10. 10.
    Henzinger, T., Horowitz, B., Majumdar, R., Wong-Toi, H.: Beyond HyTech: hybrid systems analysis using interval numerical methods. In: Lynch, N., Krogh, B. (eds.) Hybrid Systems: Computation and Control, LNCS, vol. 1790, pp. 130–144. Springer, New York (2000)Google Scholar
  11. 11.
    Ishii, D., Ueda, K., Hosobe, H.: An interval-based SAT modulo ODE solver for model checking nonlinear hybrid systems. Int. J. Softw. Tools Technol. Transf. (STTT), 1–13 (2011). doi: 10.1007/s10009-011-0193-y
  12. 12.
    Kieffer, M., Walter, E., Simeonov, I.: Guaranteed nonlinear parameter estimation for continuous-time dynamical models. In: Proceedings 14th IFAC Symposium on System Identification, Newcastle, pp. 843–848 (2006)Google Scholar
  13. 13.
    Lerch, M., Tischler, G., Gudenberg, J.W.V., Hofschuster, W., Krämer, W. Filib++, a fast interval library supporting containment computations. ACM Trans. Math. Softw. 32(2):299–324 (2006). doi: 10.1145/1141885.1141893, FILIB++ is available at http://www2.math.uni-wuppertal.de/~xsc/software/filib.html
  14. 14.
    Lygeros, J., Johansson, K., Simic, S., Zhang, J., Sastry, S.: Dynamical properties of hybrid automata. IEEE Trans. Autom. Control 48(1), 2–17 (2003). doi: 10.1109/TAC.2002.806650 Google Scholar
  15. 15.
    Müller, M.: Über das Fundamentaltheorem in der Theorie der gewöhnlichen Differentialgleichungen. Mathematische Zeitschrift 26, 619–645 (1927)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Nedialkov, N.S.: Computing rigorous bounds on the solution of an initial value problem for an ordinary differential equation. PhD thesis, Department of Computer Science, University of Toronto, Toronto, M5S 3G4 (1999)Google Scholar
  17. 17.
    Nedialkov, N.S.: VNODE-LP—a validated solver for initial value problems in ordinary differential equations. Tech. Rep. CAS-06-06-NN. Department of Computing and Software, McMaster University, Hamilton, L8S 4K1, VNODE-LP is available at http://www.cas.mcmaster.ca/~nedialk/vnodelp (2006)
  18. 18.
    Nedialkov, N.S.: Implementing a rigorous ODE solver through literate programming. In: Rauh, A., Auer, E. (eds.) Modeling, Design, and Simulation of Systems with Uncertainties. Mathematical Engineering, vol. 3, pp. 3–19. Springer, New York (2011). doi: 10.1007/978-3-642-15956-5_1
  19. 19.
    Podelski, A., Wagner, S.: Region stability proofs for hybrid systems. In: Raskin, J.F., Thiagarajan, P.S. (eds.) FORMATS, LNCS, vol. 4763, pp. 320–335. Springer, Berlin (2007)Google Scholar
  20. 20.
    Ramdani, N., Meslem, N., Candau, Y.: A hybrid bounding method for computing an over-approximation for the reachable space of uncertain nonlinear systems. IEEE Trans. Autom. Control 54(10), 2352–2364 (2009)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Ramdani, N., Meslem, N., Candau, Y.: Computing reachable sets for uncertain nonlinear monotone systems. Nonlinear Anal. Hybrid Syst. 4(2), 263–278 (2010)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Ratschan, S., She, Z.: Safety verification of hybrid systems by constraint propagation based abstraction refinement. ACM Trans. Embed. Comput. Syst. 6(1), (2007)Google Scholar
  23. 23.
    Shtrichman, O.: Tuning SAT checkers for bounded model checking. In: Emerson, E., Sistla, A. (eds.) Computer Aided Verification, LNCS, vol. 1855, pp. 480–494. Springer, Berlin (2000). doi: 10.1007/10722167_36
  24. 24.
    Stauning, O.: Automatic validation of numerical solutions. PhD thesis, Technical University of Denmark, Lyngby, (1997). http://www2.imm.dtu.dk/documents/ftp/phdliste/phd36_97.ps, FADBAD++ is available at http://www.fadbad.com
  25. 25.
    Stursberg, O., Kowalewski, S., Hoffmann, I., Preußig, J.: Comparing timed and hybrid automata as approximations of continuous systems. In: Antsakalis, P., Kohn, W., Nerode, A., Sastry, S. (eds.) Hybrid Systems IV, LNCS, vol. 1273, pp. 361–377. Springer, Berlin (1997). doi: 10.1007/bfb0031569

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Andreas Eggers
    • 1
  • Nacim Ramdani
    • 2
  • Nedialko S. Nedialkov
    • 3
  • Martin Fränzle
    • 1
  1. 1.Department of Computing ScienceCarl von Ossietzky UniversitätOldenburgGermany
  2. 2.Université d’OrléansBourgesFrance
  3. 3.McMaster UniversityHamiltonCanada

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