Efficient Verified (UN)SAT Certificate Checking

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10395)

Abstract

We present an efficient formally verified checker for satisfiability and unsatisfiability certificates for Boolean formulas in conjunctive normal form. It utilizes a two phase approach: Starting from a DRAT certificate, the unverified generator computes an enriched certificate, which is checked against the original formula by the verified checker.

Using the Isabelle/HOL Refinement Framework, we verify the actual implementation of the checker, specifying the semantics of the formula down to the integer sequence that represents it.

On a realistic benchmark suite drawn from the 2016 SAT competition, our approach is more than two times faster than the unverified standard tool drat-trim. Additionally, we implemented a multi-threaded version of the generator, which further reduces the runtime.

References

  1. 1.
    Back, R.-J.: On the correctness of refinement steps in program development. Ph.D. thesis, Department of Computer Science, University of Helsinki (1978)Google Scholar
  2. 2.
    Back, R.-J., von Wright, J.: Refinement Calculus - A Systematic Introduction. Springer, New York (1998)CrossRefMATHGoogle Scholar
  3. 3.
    Bertot, Y., Castran, P.: Interactive Theorem Proving and Program Development: Coq’Art the Calculus of Inductive Constructions, 1st edn. Springer, New York (2010)Google Scholar
  4. 4.
    Brunner, J., Lammich, P.: Formal verification of an executable LTL model checker with partial order reduction. In: Rayadurgam, S., Tkachuk, O. (eds.) NFM 2016. LNCS, vol. 9690, pp. 307–321. Springer, Cham (2016). doi:10.1007/978-3-319-40648-0_23 CrossRefGoogle Scholar
  5. 5.
    Bulwahn, L., Krauss, A., Haftmann, F., Erkök, L., Matthews, J.: Imperative functional programming with Isabelle/HOL. In: Mohamed, O.A., Muñoz, C., Tahar, S. (eds.) TPHOLs 2008. LNCS, vol. 5170, pp. 134–149. Springer, Heidelberg (2008). doi:10.1007/978-3-540-71067-7_14 CrossRefGoogle Scholar
  6. 6.
    Cruz-Filipe, L., Heule, M., Hunt, W., Matt, K., Schneider-Kamp, P.: Efficient certified RAT verification. In: de Moura, L. (ed.) CADE 2017. LNAI, vol. 10395, pp. 220–236. Springer, Cham (2017)Google Scholar
  7. 7.
    Cruz-Filipe, L., Marques-Silva, J., Schneider-Kamp, P.: Efficient certified resolution proof checking. In: Legay, A., Margaria, T. (eds.) TACAS 2017. LNCS, vol. 10205, pp. 118–135. Springer, Heidelberg (2017). doi:10.1007/978-3-662-54577-5_7 CrossRefGoogle Scholar
  8. 8.
    Darbari, A., Fischer, B., Marques-Silva, J.: Industrial-strength certified SAT solving through verified SAT proof checking. In: Cavalcanti, A., Deharbe, D., Gaudel, M.-C., Woodcock, J. (eds.) ICTAC 2010. LNCS, vol. 6255, pp. 260–274. Springer, Heidelberg (2010). doi:10.1007/978-3-642-14808-8_18 CrossRefGoogle Scholar
  9. 9.
    DRAT-TRIM GitHub repository. https://github.com/marijnheule/drat-trim
  10. 10.
  11. 11.
  12. 12.
    Esparza, J., Lammich, P., Neumann, R., Nipkow, T., Schimpf, A., Smaus, J.-G.: A fully verified executable LTL model checker. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 463–478. Springer, Heidelberg (2013). doi:10.1007/978-3-642-39799-8_31 CrossRefGoogle Scholar
  13. 13.
    Goldberg, E., Novikov, Y.: Verification of proofs of unsatisfiability for CNF formulas. In: Proceedings of DATE. IEEE (2003)Google Scholar
  14. 14.
    Gordon, M.: From LCF to HOL: a short history. In: Proof, Language, and Interaction, pp. 169–185. MIT Press (2000)Google Scholar
  15. 15.
    Haftmann, F.: Code generation from specifications in higher order logic. Ph.D. thesis, Technische Universität München (2009)Google Scholar
  16. 16.
    Haftmann, F., Krauss, A., Kunčar, O., Nipkow, T.: Data refinement in Isabelle/HOL. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 100–115. Springer, Heidelberg (2013). doi:10.1007/978-3-642-39634-2_10 CrossRefGoogle Scholar
  17. 17.
    Haftmann, F., Nipkow, T.: Code generation via higher-order rewrite systems. In: Blume, M., Kobayashi, N., Vidal, G. (eds.) FLOPS 2010. LNCS, vol. 6009, pp. 103–117. Springer, Heidelberg (2010). doi:10.1007/978-3-642-12251-4_9 CrossRefGoogle Scholar
  18. 18.
    Heule, M., Hunt, W., Wetzler, N.: Trimming while checking clausal proofs. In: 2013 Formal Methods in Computer-Aided Design, FMCAD 2013, pp. 181–188. IEEE (2013)Google Scholar
  19. 19.
    Kumar, R., Myreen, M.O., Norrish, M., Owens, S.: CakeML: a verified implementation of ML. In: Proceedings of POPL, pp. 179–192. ACM (2014)Google Scholar
  20. 20.
    Lammich, P.: Grat tool chain homepage. http://www21.in.tum.de/lammich/grat/
  21. 21.
    Lammich, P.: Gratchk proof outline. http://www21.in.tum.de/lammich/grat/outline.pdf
  22. 22.
    Lammich, P.: Automatic data refinement. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 84–99. Springer, Heidelberg (2013). doi:10.1007/978-3-642-39634-2_9 CrossRefGoogle Scholar
  23. 23.
    Lammich, P.: Verified efficient implementation of gabow’s strongly connected component algorithm. In: Klein, G., Gamboa, R. (eds.) ITP 2014. LNCS, vol. 8558, pp. 325–340. Springer, Cham (2014). doi:10.1007/978-3-319-08970-6_21 Google Scholar
  24. 24.
    Lammich, P.: Refinement to Imperative/HOL. In: Urban, C., Zhang, X. (eds.) ITP 2015. LNCS, vol. 9236, pp. 253–269. Springer, Cham (2015). doi:10.1007/978-3-319-22102-1_17 Google Scholar
  25. 25.
    Lammich, P.: Refinement based verification of imperative data structures. In: CPP, pp. 27–36. ACM (2016)Google Scholar
  26. 26.
    Lammich, P., Lochbihler, A.: The isabelle collections framework. In: Kaufmann, M., Paulson, L.C. (eds.) ITP 2010. LNCS, vol. 6172, pp. 339–354. Springer, Heidelberg (2010). doi:10.1007/978-3-642-14052-5_24 CrossRefGoogle Scholar
  27. 27.
    Lammich, P., Neumann, R.: A framework for verifying depth-first search algorithms. In: CPP 2015, pp. 137–146. ACM, New York (2015)Google Scholar
  28. 28.
    Lammich, P., Sefidgar, S.R.: Formalizing the Edmonds-Karp algorithm. In: Blanchette, J.C., Merz, S. (eds.) ITP 2016. LNCS, vol. 9807, pp. 219–234. Springer, Cham (2016). doi:10.1007/978-3-319-43144-4_14 CrossRefGoogle Scholar
  29. 29.
    Lammich, P., Tuerk, T.: Applying data refinement for monadic programs to Hopcroft’s algorithm. In: Beringer, L., Felty, A. (eds.) ITP 2012. LNCS, vol. 7406, pp. 166–182. Springer, Heidelberg (2012). doi:10.1007/978-3-642-32347-8_12 CrossRefGoogle Scholar
  30. 30.
    Milner, R., Harper, R., MacQueen, D., Tofte, M.: The Definition of Standard ML. MIT Press, Cambridge (1997)Google Scholar
  31. 31.
    MLton Standard ML compiler. http://mlton.org/
  32. 32.
    Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: engineering an efficient SAT solver. In: Proceedings of DAC, pp. 530–535. ACM (2001)Google Scholar
  33. 33.
    Myreen, M.O., Owens, S.: Proof-producing translation of higher-order logic into pure and stateful ML. J. Funct. Program. 24(2–3), 284–315 (2014)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Nipkow, T., Wenzel, M., Paulson, L.C. (eds.): Isabelle/HOL — A Proof Assistant for Higher-Order Logic. LNCS, vol. 2283. Springer, Heidelberg (2002)MATHGoogle Scholar
  35. 35.
    SAT competition (2013). http://satcompetition.org/2013/
  36. 36.
    SAT competition (2014). http://satcompetition.org/2014/
  37. 37.
    Proceedings of SAT Competition 2016: Solver and Benchmark Descriptions, vol. B-2016-1. University of Helsinki (2016)Google Scholar
  38. 38.
  39. 39.
    Sinz, C., Biere, A.: Extended resolution proofs for conjoining BDDs. In: Grigoriev, D., Harrison, J., Hirsch, E.A. (eds.) CSR 2006. LNCS, vol. 3967, pp. 600–611. Springer, Heidelberg (2006). doi:10.1007/11753728_60 CrossRefGoogle Scholar
  40. 40.
    Soos, M., Nohl, K., Castelluccia, C.: Extending SAT solvers to cryptographic problems. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 244–257. Springer, Heidelberg (2009). doi:10.1007/978-3-642-02777-2_24 CrossRefGoogle Scholar
  41. 41.
    Wetzler, N., Heule, M.J.H., Hunt, W.A.: Mechanical verification of SAT refutations with extended resolution. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 229–244. Springer, Heidelberg (2013). doi:10.1007/978-3-642-39634-2_18 CrossRefGoogle Scholar
  42. 42.
    Wetzler, N., Heule, M.J.H., Hunt, W.A.: DRAT-trim: efficient checking and trimming using expressive clausal proofs. In: Sinz, C., Egly, U. (eds.) SAT 2014. LNCS, vol. 8561, pp. 422–429. Springer, Cham (2014). doi:10.1007/978-3-319-09284-3_31 Google Scholar
  43. 43.
    Wirth, N.: Program development by stepwise refinement. Commun. ACM 14(4), 221–227 (1971)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Technische Universität MünchenMunichGermany

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