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International Workshop on Verification, Model Checking, and Abstract Interpretation

VMCAI 2012: Verification, Model Checking, and Abstract Interpretation pp 315–331Cite as

Automating Induction with an SMT Solver

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 7148)

Abstract

Mechanical proof assistants have always had support for inductive proofs. Sometimes an alternative to proof assistants, satisfiability modulo theories (SMT) solvers bring the hope of a higher degree of automation. However, SMT solvers do not natively support induction, so inductive proofs require some encoding into the SMT solver’s input.

This paper shows a surprisingly simple tactic—a rewriting strategy and a heuristic for when to apply it—that has shown to be useful in verifying simple inductive theorems, like those that can occur during program verification.

The paper describes the tactic and its implementation in a program verifier, and reports on the positive experience with using the tactic.

Keywords

  • Test Suite
  • Recursive Function
  • Recursive Call
  • Proof Obligation
  • Source Language

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Back, R.-J., von Wright, J.: Refinement Calculus: A Systematic Introduction. Graduate Texts in Computer Science. Springer, Heidelberg (1998)

    CrossRef  MATH  Google Scholar 

  2. Barnett, M., Chang, B.-Y.E., DeLine, R., Jacobs, B., Leino, K.R.M.: Boogie: A Modular Reusable Verifier for Object-Oriented Programs. In: de Boer, F.S., Bonsangue, M.M., Graf, S., de Roever, W.-P. (eds.) FMCO 2005. LNCS, vol. 4111, pp. 364–387. Springer, Heidelberg (2006)

    CrossRef  Google Scholar 

  3. Beckert, B., Hähnle, R., Schmitt, P.H. (eds.): Verification of Object-Oriented Software. LNCS (LNAI), vol. 4334. Springer, Heidelberg (2007)

    Google Scholar 

  4. Bertot, Y., Castéran, P.: Interactive Theorem Proving and Program Development, Coq’Art: The Calculus of Inductive Constructions. Texts in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg (2004)

    CrossRef  MATH  Google Scholar 

  5. Böhme, S., Nipkow, T.: Sledgehammer: Judgement day. In: Giesl, J., Hähnle, R. (eds.) IJCAR 2010. LNCS, vol. 6173, pp. 107–121. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  6. Boyer, R.S., Moore, J.S.: A Computational Logic. ACM Monograph Series. Academic Press (1979)

    Google Scholar 

  7. Bundy, A., Basin, D., Hutter, D., Ireland, A.: Rippling: Meta-level Guidance for Mathematical Reasoning. Cambridge Tracts in Theoretical Computer Science, vol. 56. Cambridge University Press (2005)

    Google Scholar 

  8. Chamarthi, H.R., Dillinger, P.C., Manolios, P., Vroon, D.: The ACL2 Sedan Theorem Proving System. In: Abdulla, P.A., Leino, K.R.M. (eds.) TACAS 2011. LNCS, vol. 6605, pp. 291–295. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  9. Cohen, E., Dahlweid, M., Hillebrand, M.A., Leinenbach, D., Moskal, M., Santen, T., Schulte, W., Tobies, S.: VCC: A Practical System for Verifying Concurrent C. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 23–42. Springer, Heidelberg (2009)

    CrossRef  Google Scholar 

  10. 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)

    CrossRef  Google Scholar 

  11. Detlefs, D., Nelson, G., Saxe, J.B.: Simplify: a theorem prover for program checking. Journal of the ACM 52(3), 365–473 (2005)

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. Filliâtre, J.-C., Marché, C.: The Why/Krakatoa/Caduceus Platform for Deductive Program Verification. In: Damm, W., Hermanns, H. (eds.) CAV 2007. LNCS, vol. 4590, pp. 173–177. Springer, Heidelberg (2007)

    CrossRef  Google Scholar 

  13. Jacobs, B., Smans, J., Philippaerts, P., Vogels, F., Penninckx, W., Piessens, F.: VeriFast: A Powerful, Sound, Predictable, Fast Verifier for C and Java. In: Bobaru, M., Havelund, K., Holzmann, G.J., Joshi, R. (eds.) NFM 2011. LNCS, vol. 6617, pp. 41–55. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  14. Jacobs, B., Smans, J., Piessens, F.: VeriFast: Imperative programs as proofs. In: VSTTE Workshop on Tools & Experiments (2010)

    Google Scholar 

  15. Johansson, M., Dixon, L., Bundy, A.: Case-Analysis for Rippling and Inductive Proof. In: Kaufmann, M., Paulson, L.C. (eds.) ITP 2010. LNCS, vol. 6172, pp. 291–306. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  16. Kaufmann, M., Manolios, P., Moore, J.S.: Computer-Aided Reasoning: An Approach. Kluwer Academic Publishers (2000)

    Google Scholar 

  17. Knuth, D.E.: The Art of Computer Programming. Seminumerical Algorithms, vol. II. Addison-Wesley (1969)

    Google Scholar 

  18. Leino, K.R.M.: This is Boogie 2. Technical report, Microsoft Research (2008)

    Google Scholar 

  19. Leino, K.R.M.: Specification and verification of object-oriented software. In: Engineering Methods and Tools for Software Safety and Security. NATO Science for Peace and Security Series D: Information and Communication Security, vol. 22, pp. 231–266. IOS Press (2009); Summer School Marktoberdorf 2008 lecture notes

    Google Scholar 

  20. Leino, K.R.M.: Dafny: An Automatic Program Verifier for Functional Correctness. In: Clarke, E.M., Voronkov, A. (eds.) LPAR-16 2010. LNCS, vol. 6355, pp. 348–370. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  21. Leino, K.R.M., Rümmer, P.: A Polymorphic Intermediate Verification Language: Design and Logical Encoding. In: Esparza, J., Majumdar, R. (eds.) TACAS 2010. LNCS, vol. 6015, pp. 312–327. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  22. Leino, K.R.M., Saxe, J.B., Stata, R.: Checking Java programs via guarded commands. In: Formal Techniques for Java Programs, Technical Report 251. Fernuniversität Hagen (1999); Also available as Technical Note 1999-002, Compaq Systems Research Center

    Google Scholar 

  23. Nieuwenhuis, R., Oliveras, A., Tinelli, C.: Solving SAT and SAT modulo theories: From an abstract Davis-Putnam-Logemann-Loveland procedure to DPLL(T). Journal of the ACM 53(6), 937–977 (2006)

    CrossRef  MathSciNet  MATH  Google Scholar 

  24. Owre, S., Rushby, J.M., Shankar, N.: PVS: A Prototype Verification System. In: Kapur, D. (ed.) CADE-11 1992. LNCS (LNAI), vol. 607, pp. 748–752. Springer, Heidelberg (1992)

    Google Scholar 

  25. Sonnex, W., Drossopoulou, S., Eisenbach, S.: Zeno: A tool for the automatic verification of algebraic properties of functional programs. Technical report, Imperial College London (2011), http://pubs.doc.ic.ac.uk/zeno/

  26. Walther, C., Schweitzer, S.: About VeriFun. In: Baader, F. (ed.) CADE-19 2003. LNCS (LNAI), vol. 2741, pp. 322–327. Springer, Heidelberg (2003)

    CrossRef  Google Scholar 

  27. Zee, K., Kuncak, V., Rinard, M.: Full functional verification of linked data structures. In: PLDI 2008, pp. 349–361. ACM (2008)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Microsoft Research, Redmond, WA, USA

    K. Rustan M. Leino

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  1. K. Rustan M. Leino
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Editors and Affiliations

  1. Swiss Federal Institute of Technology Lausanne (EPFL), IC IIF LARA INR 318, Station 14, 1015, Lausanne, Switzerland

    Viktor Kuncak

  2. Institut für Informatik, Technische Universität München, Boltzmannstr. 3, 85748, Munich, Germany

    Andrey Rybalchenko

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Leino, K.R.M. (2012). Automating Induction with an SMT Solver. In: Kuncak, V., Rybalchenko, A. (eds) Verification, Model Checking, and Abstract Interpretation. VMCAI 2012. Lecture Notes in Computer Science, vol 7148. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27940-9_21

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  • DOI: https://doi.org/10.1007/978-3-642-27940-9_21

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