International Symposium on Automated Technology for Verification and Analysis

Automated Technology for Verification and Analysis pp 48-63 | Cite as

Improving Interpolants for Linear Arithmetic

  • Ernst Althaus
  • Björn Beber
  • Joschka Kupilas
  • Christoph Scholl
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9364)

Abstract

Craig interpolation for satisfiability modulo theory formulas have come more into focus for applications of formal verification. In this paper we, introduce a method to reduce the size of linear constraints used in the description of already computed interpolant in the theory of linear arithmetic with respect to the number of linear constraints. We successfully improve interpolants by combining satisfiability modulo theory and linear programming in a local search heuristic. Our experimental results suggest a lower running time and a larger reduction compared to other methods from the literature.

Keywords

Craig-interpolation Linear arithmetic Satisfiability modulo theory Linear programming 

References

  1. 1.
    Pudlák, P.: Lower bounds for resolution and cutting plane proofs and monotone computations. J. Symbolic Logic 62(3), 981–998 (1997)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    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
  3. 3.
    Albarghouthi, A., McMillan, K.L.: Beautiful interpolants. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 313–329. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  4. 4.
    Damm, W., Dierks, H., Disch, S., Hagemann, W., Pigorsch, F., Scholl, C., Waldmann, U., Wirtz, B.: Exact and fully symbolic verification of linear hybrid automata with large discrete state spaces. Sci. Comput. Program. 77(10–11), 1122–1150 (2012)CrossRefMATHGoogle Scholar
  5. 5.
    Megiddo, N.: On the complexity of polyhedral separability. Discrete Comput. Geom. 3(1), 325–337 (1988)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Scholl, C., Pigorsch, F., Disch, S., Althaus, E.: Simple interpolants for linear arithmetic. In: Design, Automation and Test in Europe Conference and Exhibition (DATE), 2014, pp. 1–6. IEEE (2014)Google Scholar
  7. 7.
    William, C.: Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory. J. Symbolic Logic 22(03), 269–285 (1957)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    McMillan, K.L.: An interpolating theorem prover. Theoret. Comput. Sci. 345(1), 101–121 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    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
  10. 10.
    Scholl, C., Disch, S., Pigorsch, F., Kupferschmid, S.: Using an SMT solver and craig interpolation to detect and remove redundant linear constraints in representations of non-convex polyhedra. In: Proceedings of the Joint Workshops of the 6th International Workshop on Satisfiability Modulo Theories and 1st International Workshop on Bit-Precise Reasoning, pp. 18–26. ACM (2008)Google Scholar
  11. 11.
    Damm, W., Disch, S., Hungar, H., Jacobs, S., Pang, J., Pigorsch, F., Scholl, C., Waldmann, U., Wirtz, B.: Exact state set representations in the verification of linear hybrid systems with large discrete state space. In: Namjoshi, K.S., Yoneda, T., Higashino, T., Okamura, Y. (eds.) ATVA 2007. LNCS, vol. 4762, pp. 425–440. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  12. 12.
    Dutertre, B., De Moura, L.: The yices SMT solver (2006). http://yices.csl.sri.com/tool-paper.pdf
  13. 13.
    Applegate, D.L., Cook, W., Dash, S., Espinoza, D.G.: Exact solutions to linear programming problems. Oper. Res. Lett. 35(6), 693–699 (2007)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Griggio, A.: A practical approach to satisfiability modulo linear integer arithmetic. JSAT 8, 1–27 (2012)MathSciNetMATHGoogle Scholar
  15. 15.

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Ernst Althaus
    • 1
    • 2
  • Björn Beber
    • 1
  • Joschka Kupilas
    • 1
  • Christoph Scholl
    • 3
  1. 1.Max Planck Institute for InformaticsSaarbrückenGermany
  2. 2.Johannes Gutenberg UniversityMainzGermany
  3. 3.Albert-Ludwigs-UniversitätFreiburg im BreisgauGermany

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