An Interpolating Sequent Calculus for Quantifier-Free Presburger Arithmetic

  • Angelo Brillout
  • Daniel Kroening
  • Philipp Rümmer
  • Thomas Wahl
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6173)

Abstract

Craig interpolation has become a versatile tool in formal verification, for instance to generate intermediate assertions for safety analysis of programs. Interpolants are typically determined by annotating the steps of an unsatisfiability proof with partial interpolants. In this paper, we consider Craig interpolation for full quantifier-free Presburger arithmetic (QFPA), for which currently no efficient interpolation procedures are known. Closing this gap, we introduce an interpolating sequent calculus for QFPA and prove it to be sound and complete. We have extended the Princess theorem prover to generate interpolating proofs, and applied it to a large number of publicly available linear integer arithmetic benchmarks. The results indicate the robustness and efficiency of our proof-based interpolation procedure.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beyer, D., Zufferey, D., Majumdar, R.: CSIsat: Interpolation for LA+EUF. In: Gupta, A., Malik, S. (eds.) CAV 2008. LNCS, vol. 5123, pp. 304–308. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  2. 2.
    Cimatti, A., Griggio, A., Sebastiani, R.: Interpolant generation for UTVPI. In: Schmidt, R.A. (ed.) CADE 2009. LNCS, vol. 5663, pp. 167–182. Springer, Heidelberg (2009)Google Scholar
  3. 3.
    Craig, W.: Linear reasoning. a new form of the Herbrand-Gentzen theorem. The Journal of Symbolic Logic 22(3), 250–268 (1957)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Fitting, M.C.: First-Order Logic and Automated Theorem Proving, 2nd edn. Springer, Heidelberg (1996)MATHGoogle Scholar
  5. 5.
    Jain, H., Clarke, E., Grumberg, O.: Efficient interpolation for linear diophantine (dis)equations and linear modular equations. In: Gupta, A., Malik, S. (eds.) CAV 2008. LNCS, vol. 5123, pp. 254–267. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Kannan, R., Bachem, A.: Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix. SIAM J. Comput. 8(4), 499–507 (1979)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kapur, D., Majumdar, R., Zarba, C.G.: Interpolation for data structures. In: SIGSOFT ’06/FSE-14, pp. 105–116. ACM, New York (2006)CrossRefGoogle Scholar
  8. 8.
    Lynch, C., Tang, Y.: Interpolants for linear arithmetic in SMT. In: Cha, S(S.), Choi, J.-Y., Kim, M., Lee, I., Viswanathan, M. (eds.) ATVA 2008. LNCS, vol. 5311, pp. 156–170. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  9. 9.
    Maehara, S.: On the interpolation theorem of Craig. Sugaku 12, 235–237 (1960)MATHMathSciNetGoogle Scholar
  10. 10.
    McMillan, K.L.: An interpolating theorem prover. Theor. Comput. Sci. 345(1) (2005)Google Scholar
  11. 11.
    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
  12. 12.
    Presburger, M.: Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. In: Comptes Rendus du Ier congrès de Mathématiciens des Pays Slaves (1929)Google Scholar
  13. 13.
    Pugh, W.: The Omega test: a fast and practical integer programming algorithm for dependence analysis. Communications of the ACM 8, 102–114 (1992)CrossRefGoogle Scholar
  14. 14.
    Rümmer, P.: A sequent calculus for integer arithmetic with counterexample generation. In: VERIFY. CEUR, vol. 259 (2007), http://ceur-ws.org/
  15. 15.
    Rümmer, P.: A constraint sequent calculus for first-order logic with linear integer arithmetic. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS (LNAI), vol. 5330, pp. 274–289. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  16. 16.
    Rybalchenko, A., Sofronie-Stokkermans, V.: Constraint solving for interpolation. In: Cook, B., Podelski, A. (eds.) VMCAI 2007. LNCS, vol. 4349, pp. 346–362. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  17. 17.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, Chichester (1986)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Angelo Brillout
    • 1
  • Daniel Kroening
    • 2
  • Philipp Rümmer
    • 2
  • Thomas Wahl
    • 2
  1. 1.ETH ZurichSwitzerland
  2. 2.Oxford University Computing LaboratoryUnited Kingdom

Personalised recommendations