Lattice-Based Refinement in Bounded Model Checking

  • Karine Even-MendozaEmail author
  • Sepideh Asadi
  • Antti E. J. Hyvärinen
  • Hana Chockler
  • Natasha Sharygina
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11294)


In this paper we present an algorithm for bounded model-checking with SMT solvers of programs with library functions—either standard or user-defined. Typically, if the program correctness depends on the output of a library function, the model-checking process either treats this function as an uninterpreted function, or is required to use a theory under which the function in question is fully defined. The former approach leads to numerous spurious counter-examples, whereas the later faces the danger of the state-explosion problem, where the resulting formula is too large to be solved by means of modern SMT solvers.

We extend the approach of user-defined summaries and propose to represent the set of existing summaries for a given library function as a lattice of subsets of summaries, with the meet and join operations defined as intersection and union, respectively. The refinement process is then triggered by the lattice traversal, where in each node the SMT solver uses the subset of SMT summaries stored in this node to search for a satisfying assignment. The direction of the traversal is determined by the results of the concretisation of an abstract counterexample obtained at the current node. Our experimental results demonstrate that this approach allows to solve a number of instances that were previously unsolvable by the existing bounded model-checkers.



We thank Grigory Fedyukovich for helpful discussions.


  1. 1.
  2. 2.
  3. 3.
    The coq proof assistant.
  4. 4.
    Competition on software verification (SV-COMP) (2017).
  5. 5.
    Alt, L., et al.: HiFrog: SMT-based function summarization for software verification. In: Legay, A., Margaria, T. (eds.) TACAS 2017. LNCS, vol. 10206, pp. 207–213. Springer, Heidelberg (2017). Scholar
  6. 6.
    Alt, L., Fedyukovich, G., Hyvärinen, A.E.J., Sharygina, N.: A proof-sensitive approach for small propositional interpolants. In: Gurfinkel, A., Seshia, S.A. (eds.) VSTTE 2015. LNCS, vol. 9593, pp. 1–18. Springer, Cham (2016). Scholar
  7. 7.
    Alt, L., Hyvärinen, A.E.J., Sharygina, N.: LRA interpolants from no man’s land. Hardware and Software: Verification and Testing. LNCS, vol. 10629, pp. 195–210. Springer, Cham (2017). Scholar
  8. 8.
    Alt, L., Hyvärinen, A.E.J., Asadi, S., Sharygina, N.: Duality-based interpolation for quantifier-free equalities and uninterpreted functions. In: Stewart, D., Weissenbacher, G. (eds.) Proceedings of FMCAD 2017, pp. 39–46. IEEE (2017)Google Scholar
  9. 9.
    Anderson, I.: Combinatorics of Finite Sets. Clarendon Press, Oxford (1987)zbMATHGoogle Scholar
  10. 10.
    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). Scholar
  11. 11.
    Birkhoff, G.: Lattice Theory, 3rd edn. AMS, Providence (1967)zbMATHGoogle Scholar
  12. 12.
    Cimatti, A., Griggio, A., Irfan, A., Roveri, M., Sebastiani, R.: Invariant checking of NRA transition systems via incremental reduction to LRA with EUF. In: Legay, A., Margaria, T. (eds.) TACAS 2017. LNCS, vol. 10205, pp. 58–75. Springer, Heidelberg (2017). Scholar
  13. 13.
    Cimatti, A., Griggio, A., Irfan, A., Roveri, M., Sebastiani, R.: Satisfiability modulo transcendental functions via incremental linearization. In: de Moura, L. (ed.) CADE 2017. LNCS (LNAI), vol. 10395, pp. 95–113. Springer, Cham (2017). Scholar
  14. 14.
    Clarke, E., Grumberg, O., Jha, S., Lu, Y., Veith, H.: Counterexample-guided abstraction refinement. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 154–169. Springer, Heidelberg (2000). Scholar
  15. 15.
    Clarke, E., Kroening, D., Lerda, F.: A tool for checking ANSI-C programs. In: Jensen, K., Podelski, A. (eds.) TACAS 2004. LNCS, vol. 2988, pp. 168–176. Springer, Heidelberg (2004). Scholar
  16. 16.
    Clarke, E.M., Grumberg, O., Jha, S., Lu, Y., Veith, H.: Counterexample-guided abstraction refinement for symbolic model checking. J. ACM 50(5), 752–794 (2003)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Cousot, P.: Partial completeness of abstract fixpoint checking. In: Choueiry, B.Y., Walsh, T. (eds.) SARA 2000. LNCS (LNAI), vol. 1864, pp. 1–25. Springer, Heidelberg (2000). Scholar
  18. 18.
    Cousot, P., Cousot, R.: Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: Proceedings of the 4th ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages, POPL 1977, pp. 238–252. ACM, New York (1977)Google Scholar
  19. 19.
    Cousot, P., Cousot, R.: Systematic design of program analysis frameworks. In: Proceedings of the 6th ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages, POPL 1979, pp. 269–282. ACM, New York (1979)Google Scholar
  20. 20.
    Craig, W.: Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory. J. Symbolic Logic 22, 269–285 (1957)MathSciNetCrossRefGoogle Scholar
  21. 21.
    de 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). Scholar
  22. 22.
    Detlefs, D., Nelson, G., Saxe, J.B.: Simplify: a theorem prover for program checking. J. ACM 52(3), 365–473 (2005)MathSciNetCrossRefGoogle Scholar
  23. 23.
    D’Silva, V., Kroening, D., Purandare, M., Weissenbacher, G.: Interpolant strength. In: Barthe, G., Hermenegildo, M. (eds.) VMCAI 2010. LNCS, vol. 5944, pp. 129–145. Springer, Heidelberg (2010). Scholar
  24. 24.
    Ekici, B., Mebsout, A., Tinelli, C., Keller, C., Katz, G., Reynolds, A., Barrett, C.: SMTCoq: a plug-in for integrating SMT solvers into Coq. In: Majumdar, R., Kunčak, V. (eds.) CAV 2017. LNCS, vol. 10427, pp. 126–133. Springer, Cham (2017). Scholar
  25. 25.
    Giacobazzi, R., Quintarelli, E.: Incompleteness, counterexamples, and refinements in abstract model-checking. In: Cousot, P. (ed.) SAS 2001. LNCS, vol. 2126, pp. 356–373. Springer, Heidelberg (2001). Scholar
  26. 26.
    Giacobazzi, R., Ranzato, F., Scozzari, F.: Making abstract interpretations complete. J. ACM 47(2), 361–416 (2000)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Ho, Y.S., Chauhan, P., Roy, P., Mishchenko, A., Brayton, R.: Efficient uninterpreted function abstraction and refinement for word-level model checking. In: FMCAD, pp. 65–72. ACM (2016)Google Scholar
  28. 28.
    Hyvärinen, A.E.J., Asadi, S., Even-Mendoza, K., Fedyukovich, G., Chockler, H., Sharygina, N.: Theory refinement for program verification. In: Gaspers, S., Walsh, T. (eds.) SAT 2017. LNCS, vol. 10491, pp. 347–363. Springer, Cham (2017). Scholar
  29. 29.
    Hyvärinen, A.E.J., Marescotti, M., Alt, L., Sharygina, N.: OpenSMT2: an SMT solver for multi-core and cloud computing. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 547–553. Springer, Cham (2016). Scholar
  30. 30.
    Jančík, P., Alt, L., Fedyukovich, G., Hyvärinen, A.E.J., Kofroň, J., Sharygina, N.: PVAIR: partial variable assignment InterpolatoR. In: Stevens, P., Wasowski, A. (eds.) FASE 2016. LNCS, vol. 9633, pp. 419–434. Springer, Heidelberg (2016). Scholar
  31. 31.
    Kutsuna, T., Ishii, Y., Yamamoto, A.: Abstraction and refinement of mathematical functions toward smt-based test-case generation. Int. J. Softw. Tools Technol. Transfer 18(1), 109–120 (2016)CrossRefGoogle Scholar
  32. 32.
    Rollini, S.F., Alt, L., Fedyukovich, G., Hyvärinen, A.E.J., Sharygina, N.: PeRIPLO: a framework for producing effective interpolants in SAT-based software verification. In: McMillan, K., Middeldorp, A., Voronkov, A. (eds.) LPAR 2013. LNCS, vol. 8312, pp. 683–693. Springer, Heidelberg (2013). Scholar
  33. 33.
    Rummer, P., Subotic, P.: Exploring interpolants. In: Formal Methods in Computer-Aided Design (FMCAD), pp. 69–76. IEEE (2013)Google Scholar
  34. 34.
    Sery, O., Fedyukovich, G., Sharygina, N.: FunFrog: bounded model checking with interpolation-based function summarization. In: Chakraborty, S., Mukund, M. (eds.) ATVA 2012. LNCS, pp. 203–207. Springer, Heidelberg (2012). Scholar
  35. 35.
    Sery, O., Fedyukovich, G., Sharygina, N.: Interpolation-based function summaries in bounded model checking. In: Eder, K., Lourenço, J., Shehory, O. (eds.) HVC 2011. LNCS, vol. 7261, pp. 160–175. Springer, Heidelberg (2012). Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Karine Even-Mendoza
    • 1
    Email author
  • Sepideh Asadi
    • 2
  • Antti E. J. Hyvärinen
    • 2
  • Hana Chockler
    • 1
  • Natasha Sharygina
    • 2
  1. 1.King’s College LondonLondonUK
  2. 2.Università della Svizzera italianaLuganoSwitzerland

Personalised recommendations