Skip to main content
SpringerLink
Log in
Book cover

International Conference on Automated Reasoning with Analytic Tableaux and Related Methods

TABLEAUX 2011: Automated Reasoning with Analytic Tableaux and Related Methods pp 1–16Cite as

On Interpolation in Decision Procedures

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 6793))

Abstract

Interpolation means finding intermediate formulae between given formulae. When formulae decorate program locations, and describe sets of program states, interpolation may enable a program analyzer to discover information about intermediate locations and states. This mechanism has an increasing number of applications, that are relevant to program analysis and synthesis. We study interpolation in theorem proving decision procedures based on the DPLL(\({\cal T}\)) paradigm. We survey interpolation systems for DPLL, equality sharing and DPLL(\({\cal T}\)), reconstructing from the literature their completeness proofs, and clarifying the requirements for interpolation in the presence of equality.

Research supported in part by MIUR grant no. 2007-9E5KM8 and EU COST Action IC0901.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bonacina, M.P.: On theorem proving for program checking – Historical perspective and recent developments. In: Fernandez, M. (ed.) Proc. of the 12th Int. Symp. on Principles and Practice of Declarative Programming, pp. 1–11. ACM Press, New York (2010)

    Google Scholar 

  2. Bonacina, M.P., Dershowitz, N.: Abstract canonical inference. ACM Trans. on Computational Logic 8(1), 180–208 (2007)

    Article  MathSciNet  Google Scholar 

  3. Bonacina, M.P., Johansson, M.: On theorem proving with interpolation for program checking. In: Technical report, Dipartimento di Informatica. Università degli Studi di Verona (April 2011)

    Google Scholar 

  4. Bonacina, M.P., Lynch, C.A., de Moura, L.: On deciding satisfiability by theorem proving with speculative inferences. Journal of Automated Reasoning, 1–29 (in press); (Published online December 22, 2010) doi:10.1007/s10817-010-9213-y

    Google Scholar 

  5. Bradley, A.R., Manna, Z. (eds.): The Calculus of Computation – Decision Procedures with Applications to Verification. Springer, Heidelberg (2007)

    MATH  Google Scholar 

  6. Brillout, A., Kroening, D., Rümmer, P., Wahl, T.: An Interpolating Sequent Calculus for Quantifier-Free Presburger Arithmetic. In: Giesl, J., Hähnle, R. (eds.) IJCAR 2010. LNCS, vol. 6173, pp. 384–399. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  7. Brillout, A., Kroening, D., Rümmer, P., Wahl, T.: Program verification via Craig interpolation for Presburger arithmetic with arrays. Notes of the 6th Int. Verification Workshop (2010), http://www.philipp.ruemmer.org/

  8. Bruttomesso, R., Ghilardi, S., Ranise, S.: Rewriting-based quantifier-free interpolation for a theory of arrays. In: Proc. of the 22nd Int. Conf. on Rewriting Techniques and Applications, LIPICS. Leibniz-Zentrum für Informatik, Dagsthul Publishing (2011)

    Google Scholar 

  9. Bruttomesso, R., Rollini, S., Sharygina, N., Tsitovich, A.: Flexible interpolation generation in satisfiability modulo theories. In: Proc. of the 14th Int. Conf. on Computer-Aided Design, pp. 770–777. IEEE Computer Society Press, Los Alamitos (2010)

    Google Scholar 

  10. Chadha, R., Plaisted, D.A.: On the mechanical derivation of loop invariants. Journal of Symbolic Computation 15(5-6), 705–744 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cimatti, A., Griggio, A., Sebastiani, R.: Efficient Interpolant Generation in Satisfiability Modulo Theories. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 397–412. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  12. Cimatti, A., Griggio, A., Sebastiani, R.: Interpolant Generation for UTVPI. In: Schmidt, R. (ed.) CADE-22. LNCS (LNAI), vol. 5663, pp. 167–182. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  13. de Moura, L., Bjørner, N.: Model-based theory combination. In: Krstić, S., Oliveras, A. (eds.) CAV 2007. ENTCS, vol. 198(2), pp. 37–49. Elsevier, Amsterdam (2008)

    Google Scholar 

  14. D’Silva, V.: Propositional Interpolation and Abstract Interpretation. In: Gordon, A.D. (ed.) ESOP 2010. LNCS, vol. 6012, pp. 185–204. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  15. 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)

    Chapter  Google Scholar 

  16. Fuchs, A., Goel, A., Grundy, J., Krstić, S., Tinelli, C.: Ground Interpolation for the Theory of Equality. In: Kowalewski, S., Philippou, A. (eds.) TACAS 2009. LNCS, vol. 5505, pp. 413–427. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  17. Goel, A., Krstić, S., Tinelli, C.: Ground Interpolation for Combined Theories. In: Schmidt, R. (ed.) CADE-22. LNCS, vol. 5663, pp. 183–198. Springer, Heidelberg (2009)

    Google Scholar 

  18. Henzinger, T.A., Jhala, R., Majumdar, R., McMillan, K.L.: Abstractions from proofs. In: Leroy, X. (ed.) Proc. of the 31st ACM SIGACT-SIGPLAN Symp. on Principles of Programming Languages, pp. 232–244. ACM Press, New York (2004)

    Google Scholar 

  19. Hoder, K., Kovács, L., Voronkov, A.: Interpolation and Symbol Elimination in Vampire. In: Giesl, J., Hähnle, R. (eds.) IJCAR 2010. LNCS (LNAI), vol. 6173, pp. 188–195. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  20. Huang, G.: Constructing Craig interpolation formulas. In: Proc. of the 1st Annual Int. Conf. on Computing and Combinatorics, pp. 181–190. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  21. Kapur, D., Majumdar, R., Zarba, C.G.: Interpolation for data structures. In: Devambu, P. (ed.) Proc. of the 14th ACM SIGSOFT Symp. on the Foundations of Software Engineering, ACM Press, New York (2006)

    Google Scholar 

  22. Kovács, L., Voronkov, A.: Finding loop invariants for programs over arrays using a theorem prover. In: Chechik, M., Wirsing, M. (eds.) FASE 2009. LNCS, vol. 5503, pp. 470–485. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  23. Kovács, L., Voronkov, A.: Interpolation and Symbol Elimination. In: Schmidt, R. (ed.) CADE-22. LNCS (LNAI), vol. 5663, pp. 199–213. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  24. Krajíček, J.: Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic. Journal of Symbolic Logic 62(2), 457–486 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  25. McMillan, K.L.: Interpolation and SAT-Based Model Checking. In: Hunt, W.J., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 1–13. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  26. McMillan, K.L.: An interpolating theorem prover. Theoretical Computer Science 345(1), 101–121 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. 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)

    Chapter  Google Scholar 

  28. McMillan, K.L.: Lazy Annotation for Program Testing and Verification. In: Cook, B., Jackson, P., Touili, T. (eds.) CAV 2010. LNCS, vol. 6174, pp. 104–118. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  29. Nelson, G., Oppen, D.C.: Simplification by cooperating decision procedures. ACM Trans. on Programming Languages and Systems 1(2), 245–257 (1979)

    Article  MATH  Google Scholar 

  30. Nelson, G., Oppen, D.C.: Fast decision procedures based on congruence closure. Journal of the ACM 27(2), 356–364 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  31. 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)

    Article  MathSciNet  MATH  Google Scholar 

  32. Pudlàk, P.: Lower bounds for resolution and cutting plane proofs and monotone computations. Journal of Symbolic Logic 62(3), 981–998 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sebastiani, R.: Lazy satisfiability modulo theory. Journal on Satisfiability, Boolean Modelling and Computation 3, 141–224 (2006)

    MATH  Google Scholar 

  34. Shankar, N.: Automated deduction for verification. ACM Computing Surveys 41(4), 40–96 (2009)

    Article  Google Scholar 

  35. Yorsh, G., Musuvathi, M.: A Combination Method for Generating Interpolants. In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS (LNAI), vol. 3632, pp. 353–368. Springer, Heidelberg (2005); Early version in MSR-TR-2004-108 (October 2004)

    Chapter  Google Scholar 

  36. Zhang, L., Malik, S.: Validating SAT solvers using an independent resolution-based checker: practical implementations and other applications. In: Proc. of the Conf. on Design Automation and Test in Europe, pp. 10880–10885. IEEE Computer Society Press, Los Alamitos (2003)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Informatica, Università degli Studi di Verona, Strada Le Grazie 15, I-37134, Verona, Italy

    Maria Paola Bonacina & Moa Johansson

Authors
  1. Maria Paola Bonacina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Moa Johansson
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Informatik und Angewandte Mathematik, Universität Bern, Neubrückstr. 10, 3012, Bern, Switzerland

    Kai Brünnler

  2. Mathematisches Institut, Universität Bern, Sidlerstr. 5, 3012, Bern, Switzerland

    George Metcalfe

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bonacina, M.P., Johansson, M. (2011). On Interpolation in Decision Procedures. In: Brünnler, K., Metcalfe, G. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2011. Lecture Notes in Computer Science(), vol 6793. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22119-4_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-22119-4_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22118-7

  • Online ISBN: 978-3-642-22119-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation