On Solving Universally Quantified Horn Clauses

  • Nikolaj Bjørner
  • Ken McMillan
  • Andrey Rybalchenko
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7935)

Abstract

Program proving can be viewed as solving for unknown relations (such as loop invariants, procedure summaries and so on) that occur in the logical verification conditions of a program, such that the verification conditions are valid. Generic logical tools exist that can solve such problems modulo certain background theories, and therefore can be used for program analysis. Here, we extend these techniques to solve for quantified relations. This makes it possible to guide the solver by constraining the form of the proof, allowing it to converge when it otherwise would not. We show how to simulate existing abstract domains in this way, without having to directly implement program analyses or make certain heuristic choices, such as the terms and predicates that form the parameters of the abstract domain. Moreover, the approach gives the flexibility to go beyond these domains and experiment quickly with various invariant forms.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Albarghouthi, A., McMillan, K.L.: Beautiful interpolants. In: CAV (2013)Google Scholar
  2. 2.
    Alberti, F., Bruttomesso, R., Ghilardi, S., Ranise, S., Sharygina, N.: Lazy abstraction with interpolants for arrays. In: Bjørner, N., Voronkov, A. (eds.) LPAR-18 2012. LNCS, vol. 7180, pp. 46–61. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  3. 3.
    Beyene, T.A., Popeea, C., Rybalchenko, A.: Solving existentially quantified Horn clauses. In: CAV (2013)Google Scholar
  4. 4.
    Bjørner, N., McMillan, K.L., Rybalchenko, A.: Program verification as Satisfiability Modulo Theories. In: SMT (2012)Google Scholar
  5. 5.
    Cousot, P.: Verification by abstract interpretation. In: Dershowitz, N. (ed.) Verification (Manna Festschrift). LNCS, vol. 2772, pp. 243–268. Springer, Heidelberg (2004)Google Scholar
  6. 6.
    Cousot, P., Cousot, R., Logozzo, F.: A parametric segmentation functor for fully automatic and scalable array content analysis. In: POPL (2011)Google Scholar
  7. 7.
    Delzanno, G., Podelski, A.: Model Checking in CLP. In: Cleaveland, W.R. (ed.) TACAS 1999. LNCS, vol. 1579, pp. 223–239. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  8. 8.
    Detlefs, D., Nelson, G., Saxe, J.B.: Simplify: a theorem prover for program checking. J. ACM 52(3) (2005)Google Scholar
  9. 9.
    Dillig, I., Dillig, T., Aiken, A.: Fluid updates: Beyond strong vs. weak updates. In: Gordon, A.D. (ed.) ESOP 2010. LNCS, vol. 6012, pp. 246–266. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Flanagan, C.: Automatic software model checking using clp. In: Degano, P. (ed.) ESOP 2003. LNCS, vol. 2618, pp. 189–203. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  11. 11.
    Flanagan, C., Qadeer, S.: Predicate abstraction for software verification. In: POPL, pp. 191–202 (2002)Google Scholar
  12. 12.
    Graf, S., Saïdi, H.: Construction of abstract state graphs with PVS. In: Grumberg, O. (ed.) CAV 1997. LNCS, vol. 1254, pp. 72–83. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  13. 13.
    Grebenshchikov, S., Lopes, N.P., Popeea, C., Rybalchenko, A.: Synthesizing software verifiers from proof rules. In: PLDI (2012)Google Scholar
  14. 14.
    Gulwani, S., McCloskey, B., Tiwari, A.: Lifting abstract interpreters to quantified logical domains. In: Necula, G.C., Wadler, P. (eds.) POPL, pp. 235–246. ACM (2008)Google Scholar
  15. 15.
    Gupta, A., Popeea, C., Rybalchenko, A.: Solving recursion-free Horn clauses over LI+UIF. In: Yang, H. (ed.) APLAS 2011. LNCS, vol. 7078, pp. 188–203. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  16. 16.
    Hermenegildo, M., Puebla, G., Bueno, F., López-García, P.: Program development using abstract interpretation (and the ciao system preprocessor). In: Cousot, R. (ed.) SAS 2003. LNCS, vol. 2694, pp. 127–152. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  17. 17.
    Hoder, K., Bjørner, N.: Generalized property directed reachability. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 157–171. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  18. 18.
    Hoder, K., Kovács, L., Voronkov, A.: Case studies on invariant generation using a saturation theorem prover. In: Batyrshin, I., Sidorov, G. (eds.) MICAI 2011, Part I. LNCS, vol. 7094, pp. 1–15. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  19. 19.
    Hojjat, H., Konečný, F., Garnier, F., Iosif, R., Kuncak, V., Rümmer, P.: A verification toolkit for numerical transition systems - tool paper. In: Giannakopoulou, D., Méry, D. (eds.) FM 2012. LNCS, vol. 7436, pp. 247–251. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  20. 20.
    Jaffar, J., Maher, M.J.: Constraint logic programming: A survey. J. Log. Program. 19(20), 503–581 (1994)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Jaffar, J., Murali, V., Navas, J.A., Santosa, A.E.: Tracer: A symbolic execution tool for verification. In: Madhusudan, P., Seshia, S.A. (eds.) CAV 2012. LNCS, vol. 7358, pp. 758–766. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  22. 22.
    Jaffar, J., Santosa, A.E., Voicu, R.: Modeling Systems in CLP. In: Gabbrielli, M., Gupta, G. (eds.) ICLP 2005. LNCS, vol. 3668, pp. 412–413. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  23. 23.
    Jhala, R., McMillan, K.L.: Array abstractions from proofs. In: Damm, W., Hermanns, H. (eds.) CAV 2007. LNCS, vol. 4590, pp. 193–206. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  24. 24.
    Komuravelli, A., Gurfinkel, A., Chaki, S., Clarke, E.: Automatic Abstraction in SMT-Based Unbounded Software Model Checking. In: CAV (2013)Google Scholar
  25. 25.
    Larraz, D., Rodríguez-Carbonell, E., Rubio, A.: SMT-Based Array Invariant Generation. In: Giacobazzi, R., Berdine, J., Mastroeni, I. (eds.) VMCAI 2013. LNCS, vol. 7737, pp. 169–188. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  26. 26.
    McMillan, K.L.: An interpolating theorem prover. Theor. Comput. Sci. 345(1) (2005)Google Scholar
  27. 27.
    McMillan, K.L.: Lazy abstraction with interpolants. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 123–136. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  28. 28.
    McMillan, K.L.: Quantified invariant generation using an interpolating saturation prover. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 413–427. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  29. 29.
    McMillan, K.L.: Interpolants from Z3 proofs. In: FMCAD (2011)Google Scholar
  30. 30.
    McMillan, K.L., Rybalchenko, A.: Computing relational fixed points using interpolation. Technical Report MSR-TR-2013-6, Microsoft Research (2013), http://research.microsoft.com/apps/pubs/?id=180055
  31. 31.
    Pnueli, A., Ruah, S., Zuck, L.D.: Automatic deductive verification with invisible invariants. In: Margaria, T., Yi, W. (eds.) TACAS 2001. LNCS, vol. 2031, pp. 82–97. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  32. 32.
    Rümmer, P., Hojjat, H., Kuncak, V.: Disjunctive interpolants for horn-clause verification. In: CAV (2013)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Nikolaj Bjørner
    • 1
  • Ken McMillan
    • 1
  • Andrey Rybalchenko
    • 1
    • 2
  1. 1.Microsoft ResearchUSA
  2. 2.Technische Universität MünchenGermany

Personalised recommendations