Challenges in Constraint-Based Analysis of Hybrid Systems

  • Andreas Eggers
  • Natalia Kalinnik
  • Stefan Kupferschmid
  • Tino Teige
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5655)


In the analysis of hybrid discrete-continuous systems, rich arithmetic constraint formulae with complex Boolean structure arise naturally. The iSAT algorithm, a solver for such formulae, is aimed at bounded model checking of hybrid systems. In this paper, we identify challenges emerging from planned and ongoing work to enhance the iSAT algorithm. First, we propose an extension of iSAT to directly handle ordinary differential equations as constraints. Second, we outline the recently introduced generalization of the iSAT algorithm to deal with probabilistic hybrid systems and some open research issues in that context. Third, we present ideas on how to move from bounded to unbounded model checking by using the concept of interpolation. Finally, we discuss the adaption of some parallelization techniques to the iSAT case, which will hopefully lead to performance gains in the future. By presenting these open research questions, this paper aims at fostering discussions on these extensions of constraint solving.


mixed Boolean and arithmetic constraints differential equations stochastic SMT Craig interpolation parallel solver 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Fränzle, M.: Analysis of hybrid systems: An ounce of realism can save an infinity of states. In: Flum, J., Rodríguez-Artalejo, M. (eds.) CSL 1999. LNCS, vol. 1683, pp. 126–140. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  2. 2.
    Henzinger, T.A., Kopke, P.W., Puri, A., Varaiya, P.: What’s decidable about hybrid automata? In: Proc. of the 27th Annual Symposium on Theory of Computing, pp. 373–382. ACM Press, New York (1995)Google Scholar
  3. 3.
    Groote, J.F., Koorn, J.W.C., van Vlijmen, S.F.M.: The Safety Guaranteeing System at Station Hoorn-Kersenboogerd. In: Conference on Computer Assurance, pp. 57–68. National Institute of Standards and Technology (1995)Google Scholar
  4. 4.
    Biere, A., Cimatti, A., Zhu, Y.: Symbolic model checking without BDDs. In: Cleaveland, W.R. (ed.) TACAS 1999. LNCS, vol. 1579, p. 193. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  5. 5.
    Audemard, G., Bozzano, M., Cimatti, A., Sebastiani, R.: Verifying industrial hybrid systems with MathSAT. In: Bounded Model Checking (BMC 2004). ENTCS, vol. 119, pp. 17–32 (2005)Google Scholar
  6. 6.
    Fränzle, M., Herde, C.: HySAT: An efficient proof engine for bounded model checking of hybrid systems. Formal Methods in System Design 30(3), 179–198 (2007)CrossRefzbMATHGoogle Scholar
  7. 7.
    Benhamou, F., Granvilliers, L.: Continuous and interval constraints. In: Rossi, F., van Beek, P., Walsh, T. (eds.) Handbook of Constraint Programming. Foundations of Artificial Intelligence, pp. 571–603. Elsevier, Amsterdam (2006)CrossRefGoogle Scholar
  8. 8.
    Barrett, C., Sebastiani, R., Seshia, S., Tinelli, C.: Satisfiability Modulo Theories. In: Handbook on Satisfiability, February 2009. Frontiers in Artificial Intelligence and Applications, vol. 185. IO Press (2009),
  9. 9.
    Bauer, A., Pister, M., Tautschnig, M.: Tool-support for the analysis of hybrid systems and models. In: Design, Automation and Test in Europe. IEEE, Los Alamitos (2007)Google Scholar
  10. 10.
    Fränzle, M., Herde, C., Teige, T., Ratschan, S., Schubert, T.: Efficient Solving of Large Non-linear Arithmetic Constraint Systems with Complex Boolean Structure. JSAT Special Issue on SAT/CP Integration 1, 209–236 (2007)zbMATHGoogle Scholar
  11. 11.
    Tseitin, G.: On the complexity of derivations in propositional calculus. Studies in Constructive Mathematics and Mathematical Logics (1968)Google Scholar
  12. 12.
    Davis, M., Putnam, H.: A Computing Procedure for Quantification Theory. Journal of the ACM 7(3), 201–215 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Davis, M., Logemann, G., Loveland, D.: A Machine Program for Theorem Proving. Communications of the ACM 5, 394–397 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Moore, R.E.: Automatic local coordinate transformation to reduce the growth of error bounds in interval computation of solutions of ordinary differential equations. In: Ball, L.B. (ed.) Error in Digital Computation, vol. II, pp. 103–140. Wiley, New York (1965)Google Scholar
  15. 15.
    Lohner, R.: Einschließung der Lösung gewöhnlicher Anfangs- und Randwertaufgaben. PhD thesis, Fakultät für Mathematik der Universität Karlsruhe, Karlsruhe (1988)Google Scholar
  16. 16.
    Stauning, O.: Automatic Validation of Numerical Solutions. PhD thesis, Technical University of Denmark, Lyngby (1997)Google Scholar
  17. 17.
    Berz, M., Makino, K.: Verified Integration of ODEs and Flows Using Differential Algebraic Methods on High-Order Taylor Models. Reliable Computing 4(4), 361–369 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Henzinger, T.A., Horowitz, B., Majumdar, R., Wong-Toi, H.: Beyond HYTECH: Hybrid systems analysis using interval numerical methods. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 130–144. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  19. 19.
    Hickey, T., Wittenberg, D.: Rigorous modeling of hybrid systems using interval arithmetic constraints. In: Alur, R., Pappas, G.J. (eds.) HSCC 2004. LNCS, vol. 2993, pp. 402–416. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  20. 20.
    Eggers, A., Fränzle, M., Herde, C.: SAT modulo ODE: A direct SAT approach to hybrid systems. In: Cha, S(S.), Choi, J.-Y., Kim, M., Lee, I., Viswanathan, M. (eds.) ATVA 2008. LNCS, vol. 5311, pp. 171–185. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  21. 21.
    Fränzle, M., Hermanns, H., Teige, T.: Stochastic Satisfiability Modulo Theory: A Novel Technique for the Analysis of Probabilistic Hybrid Systems. In: Egerstedt, M., Mishra, B. (eds.) HSCC 2008. LNCS, vol. 4981, pp. 172–186. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  22. 22.
    Papadimitriou, C.H.: Games against nature. J. Comput. Syst. Sci. 31(2), 288–301 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Schmitt, C.: Bounded Model Checking of Probabilistic Hybrid Automata. Master’s thesis, Carl von Ossietzky University, Dpt. of Computing Science, Oldenburg, Germany (March 2008)Google Scholar
  24. 24.
    Teige, T., Fränzle, M.: Stochastic Satisfiability modulo Theories for Non-linear Arithmetic. In: Perron, L., Trick, M.A. (eds.) CPAIOR 2008. LNCS, vol. 5015, pp. 248–262. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  25. 25.
    Craig, W.: Linear reasoning: A new form of the Herbrand-Gentzen theorem. Journal of Symbolic Logic 22(3), 250–268 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    McMillan, K.L.: Interpolation and SAT-based model checking. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 1–13. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  27. 27.
    McMillan, K.L.: An interpolating theorem prover. Theor. Comput. Sci. 345(1), 101–121 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Beyer, D., Henzinger, T.A., Jhala, R., Majumdar, R.: The software model checker Blast: Applications to software engineering. International Journal on Software Tools for Technology Transfer (STTT) 9(5-6), 505–525 (2007) (invited to special issue of selected papers from FASE 2004/2005)CrossRefGoogle Scholar
  29. 29.
    Rybalchenko, A., Sofronie-Stokkermans, V.: Constraint solving for interpolation. In: Cook, B., Podelski, A. (eds.) VMCAI 2007. LNCS, vol. 4349, pp. 346–362. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  30. 30.
    Cimatti, A., Griggio, A., Sebastiani, R.: Efficient interpolant generation in satisfiability modulo theories. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 397–412. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  31. 31.
    Pudlák, P.: Lower Bounds for Resolution and Cutting Plane Proofs and Monotone Computations. Journal of Symbolic Logic 62(3), 981–998 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Böhm, M., Speckenmeyer, E.: A fast parallel SAT-solver - efficient workload balancing. Annals of Mathematics and Artificial Intelligence 17(3-4), 381–400 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Sinz, C., Blochinger, W., Küchlin, W.: PaSAT - parallel SAT-checking with lemma exchange: Implementation and applications. In: Kautz, H., Selman, B. (eds.) LICS 2001 Workshop on Theory and Applications of Satisfiability Testing (SAT 2001), June 2001. Electronic Notes in Discrete Mathematics, vol. 9. Elsevier Science Publishers, Boston (2001)Google Scholar
  34. 34.
    Schubert, T., Lewis, M., Becker, B.: PaMira – A Parallel SAT Solver with Knowledge Sharing. In: 6th International Workshop on Microprocessor Test and Verification (MTV 2005), pp. 29–36. IEEE Computer Society, Los Alamitos (2005)CrossRefGoogle Scholar
  35. 35.
    Jurkowiak, B., Li, C.M., Utard, G.: Parallelizing SATZ Using Dynamic Workload Balancing. In: Proceedings of the Workshop on Theory and Applications of Satisfiability Testing (SAT 2001), June 2001, vol. 9. Elsevier Science Publishers, Amsterdam (2001)Google Scholar
  36. 36.
    Feldman, Y., Dershowitz, N., Hanna, Z.: Parallel multithreaded satisfiability solver: Design and implementation. Electronic Notes in Theoretical Computer Science 128(3), 75–90 (2005)CrossRefGoogle Scholar
  37. 37.
    Lewis, M.D.T., Schubert, T., Becker, B.: Multithreaded SAT solving. In: Proceedings of the 12th Asia and South Pacific Design Automation Conference, pp. 926–931. IEEE Computer Society, Los Alamitos (2007)Google Scholar
  38. 38.
    Zhang, H., Bonacina, M.P., Hsiang, J.: PSATO: A distributed propositional prover and its application to quasigroup problems. Journal of Symbolic Computation 21(4), 543–560 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Ábrahám, E., Schubert, T., Becker, B., Fränzle, M., Herde, C.: Parallel SAT solving in bounded model checking. In: Brim, L., Haverkort, B.R., Leucker, M., van de Pol, J. (eds.) FMICS 2006 and PDMC 2006. LNCS, vol. 4346, pp. 301–315. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  40. 40.
    Strichman, O.: Accelerating bounded model checking of safety properties. Formal Methods in System Design 24(1), 5–24 (2004)CrossRefzbMATHGoogle Scholar
  41. 41.
    Walsh, T.: Stochastic constraint programming. In: Proc. of the 15th European Conference on Artificial Intelligence (ECAI 2002). IOS Press, Amsterdam (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Andreas Eggers
    • 1
  • Natalia Kalinnik
    • 2
  • Stefan Kupferschmid
    • 2
  • Tino Teige
    • 1
  1. 1.Carl von Ossietzky Universität OldenburgGermany
  2. 2.Albert-Ludwigs-Universität FreiburgGermany

Personalised recommendations