Skip to main content
SpringerLink
Log in

Interpolation and Model Checking

  • Chapter
  • First Online:
Handbook of Model Checking

Abstract

In this chapter we consider applications of logical proofs in model checking. Here we are not concerned with using model checking to verify steps in a larger proof but rather with ways in which logical proof methods can aid model checking, particularly in focusing model-checking methods on relevant facts. We introduce a framework for abstraction refinement based on a concept of deductive generalization. We then show how various abstraction refinement schemes can be understood in this framework in terms of local proofs and Craig interpolation. This unifying view exposes the trade-offs made in different systems between the quality and cost of refinements, and also leads to novel model-checking approaches.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ball, T., Majumdar, R., Millstein, T., Rajamani, S.K.: Automatic predicate abstraction of C programs. SIGPLAN Not. 36(5), 203–213 (2001)

    Article  Google Scholar 

  2. Ball, T., Podelski, A., Rajamani, S.K.: Boolean and Cartesian abstraction for model checking C programs. Int. J. Softw. Tools Technol. Transf. 5(1), 49–58 (2003)

    Article  Google Scholar 

  3. Ball, T., Rajamani, S.K.: Bebop: a symbolic model checker for boolean programs. In: Havelund, K., Penix, J., Visser, W. (eds.) Intl. Workshop on Model Checking Software (SPIN). LNCS, vol. 1885, pp. 113–130. Springer, Heidelberg (2000)

    Google Scholar 

  4. Ball, T., Rajamani, S.K.: Generating abstract explanations of spurious counterexamples in C programs. Tech. Rep. MSR-TR-2002-09, Microsoft Research (2002)

    Google Scholar 

  5. Barrett, C.W., Sebastiani, R., Seshia, S.A., Tinelli, C.: Satisfiability modulo theories. In: Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185, pp. 825–885. IOS Press, Amsterdam (2009)

    Google Scholar 

  6. Biere, A., Kroening, D.: SAT-based model checking. In: Handbook of Model Checking. Springer, Heidelberg (2018)

    Google Scholar 

  7. Brillout, A., Kroening, D., Rümmer, P., Wahl, T.: An interpolating sequent calculus for quantifier-free Presburger arithmetic. In: Giesl, J., Hähnle, R. (eds.) Intl. Joint Conf. on Automated Reasoning (IJCAR). LNCS, vol. 6173, pp. 384–399. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  8. Brillout, A., Kroening, D., Rümmer, P., Wahl, T.: Beyond quantifier-free interpolation in extensions of Presburger arithmetic. In: Jhala, R., Schmidt, D.A. (eds.) Intl. Conf. on Verification, Model Checking and Abstract Interpretation (VMCAI). LNCS, vol. 6538, pp. 88–102. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  9. Bruttomesso, R., Ghilardi, S., Ranise, S.: Rewriting-based quantifier-free interpolation for a theory of arrays. In: Schmidt-Schauß, M. (ed.) Intl. Conf. Rewriting Techniques and Applications (RTA). LIPIcs, vol. 10, pp. 171–186. Schloss Dagstuhl—Leibniz-Zentrum fuer Informatik, Wadern (2011)

    MATH  Google Scholar 

  10. Cimatti, A., Griggio, A., Sebastiani, R.: Interpolant generation for UTVPI. In: Schmidt, R.A. (ed.) Intl. Conf. on Automated Deduction (CADE). LNCS, vol. 5663, pp. 167–182. Springer, Heidelberg (2009)

    Google Scholar 

  11. Clarke, E.M., Kroening, D., Sharygina, N., Yorav, K.: SATABS: SAT-based predicate abstraction for ANSI-C. In: Halbwachs, N., Zuck, L.D. (eds.) Intl. Conf. on Tools and Algorithms for the Construction and Analysis of Systems (TACAS). LNCS, vol. 3440, pp. 570–574. Springer, Heidelberg (2005)

    MATH  Google Scholar 

  12. Craig, W.: Linear reasoning: a new form of the Herbrand-Gentzen theorem. J. Symb. Log. 22(3), 250–268 (1957)

    Article  MathSciNet  Google Scholar 

  13. Dams, D., Grumberg, O.: Abstraction and abstraction refinement. In: Clarke, E.M., Henzinger, T.A., Veith, H., Bloem, R. (eds.) Handbook of Model Checking. Springer, Heidelberg (2018)

    Google Scholar 

  14. D’Silva, V., Kroening, D., Purandare, M., Weissenbacher, G.: Interpolant strength. In: Barthe, G., Hermenegildo, M.V. (eds.) Intl. Conf. on Verification, Model Checking and Abstract Interpretation (VMCAI). LNCS, vol. 5944, pp. 129–145. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  15. Griggio, A.: Effective word-level interpolation for software verification. In: Bjesse, P., Slobodová, A. (eds.) Formal Methods in Computer Aided Design (FMCAD), pp. 28–36. FMCAD, Austin (2011)

    Google Scholar 

  16. Griggio, A., Le, T.T.H., Sebastiani, R.: Efficient interpolant generation in satisfiability modulo linear integer arithmetic. In: Abdulla, P.A., Leino, K.R.M. (eds.) Intl. Conf. on Tools and Algorithms for the Construction and Analysis of Systems (TACAS). LNCS, vol. 6605, pp. 143–157. Springer, Heidelberg (2011)

    Google Scholar 

  17. Gulavani, B.S., Henzinger, T.A., Kannan, Y., Nori, A.V., Rajamani, S.K.: SYNERGY: a new algorithm for property checking. In: Young, M., Devanbu, P.T. (eds.) Intl. Symp. on Foundations of Software Engineering (FSE), pp. 117–127. ACM, New York (2006)

    Google Scholar 

  18. Henzinger, T.A., Jhala, R., Majumdar, R., McMillan, K.L.: Abstractions from proofs. In: Jones, N.D., Leroy, X. (eds.) Symp. on Principles of Programming Languages (POPL), pp. 232–244. ACM, New York (2004)

    Google Scholar 

  19. Hoder, K., Kovács, L., Voronkov, A.: Playing in the grey area of proofs. In: Field, J., Hicks, M. (eds.) Symp. on Principles of Programming Languages (POPL), pp. 259–272. ACM, New York (2012)

    MATH  Google Scholar 

  20. Jain, H., Clarke, E.M., Grumberg, O.: Efficient Craig interpolation for linear diophantine (dis)equations and linear modular equations. In: Gupta, A., Malik, S. (eds.) Intl. Conf. on Computer-Aided Verification (CAV). LNCS, vol. 5123, pp. 254–267. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  21. Jhala, R., McMillan, K.L.: A practical and complete approach to predicate refinement. In: Hermanns, H., Palsberg, J. (eds.) Intl. Conf. on Tools and Algorithms for the Construction and Analysis of Systems (TACAS). LNCS, vol. 3920, pp. 459–473. Springer, Heidelberg (2006)

    MATH  Google Scholar 

  22. Jhala, R., Podelski, A., Rybalchenko, A.: Predicate abstraction for program verification. In: Clarke, E.M., Henzinger, T.A., Veith, H., Bloem, R. (eds.) Handbook of Model Checking. Springer, Heidelberg (2018)

    Google Scholar 

  23. Kapur, D., Majumdar, R., Zarba, C.G.: Interpolation for data structures. In: Young, M., Devanbu, P.T. (eds.) Intl. Symp. on Foundations of Software Engineering (FSE), pp. 105–116. ACM, New York (2006)

    Google Scholar 

  24. Kovács, L., Voronkov, A.: Interpolation and symbol elimination. In: Schmidt, R.A. (ed.) Intl. Conf. on Automated Deduction (CADE). LNCS, vol. 5663, pp. 199–213. Springer, Heidelberg (2009)

    Google Scholar 

  25. Kroening, D., Leroux, J., Rümmer, P.: Interpolating quantifier-free Presburger arithmetic. In: Fermüller, C.G., Voronkov, A. (eds.) Intl. Conf. on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR). LNCS, vol. 6397, pp. 489–503. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  26. Kroening, D., Sharygina, N.: Interactive presentation: image computation and predicate refinement for RTL verilog using word level proofs. In: Lauwereins, R., Madsen, J. (eds.) Design, Automation & Test in Europe (DATE), pp. 1325–1330. ACM, New York (2007)

    Google Scholar 

  27. Kroening, D., Weissenbacher, G.: Lifting propositional interpolants to the word-level. In: Formal Methods in Computer Aided Design (FMCAD), pp. 85–89. IEEE, Piscataway (2007)

    Chapter  Google Scholar 

  28. McMillan, K.L.: Interpolation and SAT-based model checking. In: Hunt, W.A., Somenzi, F. (eds.) Intl. Conf. on Computer-Aided Verification (CAV). LNCS, vol. 2725, pp. 1–13. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  29. McMillan, K.L.: An interpolating theorem prover. Theor. Comput. Sci. 345(1), 101–121 (2005)

    Article  MathSciNet  Google Scholar 

  30. McMillan, K.L.: Lazy abstraction with interpolants. In: Ball, T., Jones, R.B. (eds.) Intl. Conf. on Computer-Aided Verification (CAV). LNCS, vol. 4144, pp. 123–136. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  31. McMillan, K.L.: Quantified invariant generation using an interpolating saturation prover. In: Ramakrishnan, C.R., Rehof, J. (eds.) Intl. Conf. on Tools and Algorithms for the Construction and Analysis of Systems (TACAS). LNCS, vol. 4963, pp. 413–427. Springer, Heidelberg (2008)

    Google Scholar 

  32. McMillan, K.L.: Lazy annotation for program testing and verification. In: Touili, T., Cook, B., Jackson, P. (eds.) Intl. Conf. on Computer-Aided Verification (CAV). LNCS, vol. 6174, pp. 104–118. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  33. McMillan, K.L.: Interpolants from Z3 proofs. In: Bjesse, P., Slobodová, A. (eds.) Formal Methods in Computer Aided Design (FMCAD), pp. 19–27. FMCAD, Austin (2011)

    Google Scholar 

  34. McMillan, K.L., Amla, N.: Automatic abstraction without counterexamples. In: Garavel, H., Hatcliff, J. (eds.) Intl. Conf. on Tools and Algorithms for the Construction and Analysis of Systems (TACAS). LNCS, vol. 2619, pp. 2–17. Springer, Heidelberg (2003)

    Google Scholar 

  35. Nori, A.V., Rajamani, S.K., Tetali, S., Thakur, A.V.: The Yogi project: software property checking via static analysis and testing. In: Kowalewski, S., Philippou, A. (eds.) Intl. Conf. on Tools and Algorithms for the Construction and Analysis of Systems (TACAS). LNCS, vol. 5505, pp. 178–181. Springer, Heidelberg (2009)

    Google Scholar 

  36. Pudlák, P.: Lower bounds for resolution and cutting plane proofs and monotone computations. J. Symb. Log. 62(3), 981–998 (1997)

    Article  MathSciNet  Google Scholar 

  37. Rybalchenko, A., Sofronie-Stokkermans, V.: Constraint solving for interpolation. J. Symb. Comput. 45(11), 1212–1233 (2010)

    Article  MathSciNet  Google Scholar 

  38. Shankar, N.: Combining model checking and deduction. In: Clarke, E.M., Henzinger, T.A., Veith, H., Bloem, R. (eds.) Handbook of Model Checking. Springer, Heidelberg (2018)

    Google Scholar 

  39. Vizel, Y., Grumberg, O.: Interpolation-sequence based model checking. In: Formal Methods in Computer Aided Design (FMCAD), pp. 1–8. IEEE, Piscataway (2009)

    Google Scholar 

  40. Yorsh, G., Musuvathi, M.: A combination method for generating interpolants. In: Nieuwenhuis, R. (ed.) Intl. Conf. on Automated Deduction (CADE). LNCS, vol. 3632, pp. 353–368. Springer, Heidelberg (2005)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Microsoft Research, Redmond, WA, USA

    Kenneth L. McMillan

Authors
  1. Kenneth L. McMillan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kenneth L. McMillan .

Editor information

Editors and Affiliations

  1. FORE Systems University Professor of Computer Science, Carnegie Mellon University, Pittsburgh, Pennsylvania, USA

    Edmund M. Clarke

  2. Institute of Science and Technology Austria (IST), Klosterneuburg, Austria

    Thomas A. Henzinger

  3. Arbeitsbereich Formal Methods in Systems Engineering, Technische Universität Wien, Wien, Austria

    Helmut Veith

  4. Institut f. Angewandte Informationsverarbeitung und Kommunikationstechnologie (IAIK), Technische Universität Graz, Graz, Austria

    Roderick Bloem

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

McMillan, K.L. (2018). Interpolation and Model Checking. In: Clarke, E., Henzinger, T., Veith, H., Bloem, R. (eds) Handbook of Model Checking. Springer, Cham. https://doi.org/10.1007/978-3-319-10575-8_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-10575-8_14

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-10574-1

  • Online ISBN: 978-3-319-10575-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation