Sets with Cardinality Constraints in Satisfiability Modulo Theories

  • Philippe Suter
  • Robin Steiger
  • Viktor Kuncak
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6538)

Abstract

Boolean Algebra with Presburger Arithmetic (BAPA) is a decidable logic that can express constraints on sets of elements and their cardinalities. Problems from verification of complex properties of software often contain fragments that belong to quantifier-free BAPA (QFBAPA). In contrast to many other NP-complete problems (such as quantifier-free first-order logic or linear arithmetic), the applications of QFBAPA to a broader set of problems has so far been hindered by the lack of an efficient implementation that can be used alongside other efficient decision procedures. We overcome these limitations by extending the efficient SMT solver Z3 with the ability to reason about cardinality (QFBAPA) constraints. Our implementation uses the DPLL(T) mechanism of Z3 to reason about the top-level propositional structure of a QFBAPA formula, improving the efficiency compared to previous implementations. Moreover, we present a new algorithm for automatically decomposing QFBAPA formulas. Our algorithm alleviates the exponential explosion of considering all Venn regions, significantly improving the tractability of formulas with many set variables. Because it is implemented as a theory plugin, our implementation enables Z3 to prove formulas that use QFBAPA constructs with constructs from other theories that Z3 supports, as well as with quantifiers. We have applied our implementation to the verification of functional programs; we show it can automatically prove formulas that no automated approach was reported to be able to prove before.

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References

  1. 1.
    Beyer, D., Henzinger, T.A., Jhala, R., Majumdar, R.: The software model checker Blast. STTT 9(5-6), 505–525 (2007)CrossRefGoogle Scholar
  2. 2.
    Dewar, R.K.: Programming by refinement, as exemplified by the SETL representation sublanguage. ACM TOPLAS (July 1979)Google Scholar
  3. 3.
    Feferman, S., Vaught, R.L.: The first order properties of products of algebraic systems. Fundamenta Mathematicae 47, 57–103 (1959)MathSciNetMATHGoogle Scholar
  4. 4.
    Gottlob, G., Greco, G., Marnette, B.: HyperConsistency width for constraint satisfaction: Algorithms and complexity results. In: Lipshteyn, M., Levit, V.E., McConnell, R.M. (eds.) Graph Theory, Computational Intelligence and Thought. LNCS, vol. 5420, pp. 87–99. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  5. 5.
    Gulwani, S., Lev-Ami, T., Sagiv, M.: A combination framework for tracking partition sizes. In: POPL, pp. 239–251 (2009)Google Scholar
  6. 6.
    Krstić, S., Goel, A., Grundy, J., Tinelli, C.: Combined satisfiability modulo parametric theories. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 602–617. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. 7.
    Kuncak, V.: Modular Data Structure Verification. Ph.D. thesis, EECS Department, Massachusetts Institute of Technology (February 2007)Google Scholar
  8. 8.
    Kuncak, V., Nguyen, H.H., Rinard, M.: Deciding Boolean Algebra with Presburger Arithmetic. J. of Automated Reasoning (2006)Google Scholar
  9. 9.
    Kuncak, V., Piskac, R., Suter, P., Wies, T.: Building a calculus of data structures. In: Barthe, G., Hermenegildo, M. (eds.) VMCAI 2010. LNCS, vol. 5944, pp. 26–44. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Kuncak, V., Rinard, M.: Towards Efficient Satisfiability Checking for Boolean Algebra with Presburger Arithmetic. In: Pfenning, F. (ed.) CADE 2007. LNCS (LNAI), vol. 4603, pp. 215–230. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  11. 11.
    Lam, P., Kuncak, V., Rinard, M.: Generalized typestate checking for data structure consistency. In: Cousot, R. (ed.) VMCAI 2005. LNCS, vol. 3385, pp. 430–447. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  12. 12.
    Liang, S.: The Java Native Interface: Programmer’s Guide and Specification. Addison-Wesley, Reading (1999)Google Scholar
  13. 13.
    de Moura, L., Bjørner, N.: Model-based theory combination. Electronic Notes in Theoretical Computer Science 198(2), 37–49 (2008)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    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)CrossRefGoogle Scholar
  15. 15.
    Pérez, J.A.N., Rybalchenko, A., Singh, A.: Cardinality abstraction for declarative networking applications. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 584–598. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  16. 16.
    Suter, P., Dotta, M., Kuncak, V.: Decision procedures for algebraic data types with abstractions. In: POPL (2010)Google Scholar
  17. 17.
    Wies, T., Piskac, R., Kuncak, V.: Combining theories with shared set operations. In: Ghilardi, S., Sebastiani, R. (eds.) FroCoS 2009. LNCS, vol. 5749, pp. 366–382. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  18. 18.
    Yessenov, K., Kuncak, V., Piskac, R.: Collections, cardinalities, and relations. In: Barthe, G., Hermenegildo, M. (eds.) VMCAI 2010. LNCS, vol. 5944, pp. 380–395. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  19. 19.
    Zarba, C.G.: Combining sets with integers. In: Armando, A. (ed.) FroCos 2002. LNCS (LNAI), vol. 2309, pp. 103–116. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  20. 20.
    Zee, K., Kuncak, V., Rinard, M.: Full functional verification of linked data structures. In: PLDI, pp. 349–361 (2008)Google Scholar
  21. 21.
    Zee, K., Kuncak, V., Rinard, M.: An integrated proof language for imperative programs. In: PLDI, pp. 338–351 (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Philippe Suter
    • 1
  • Robin Steiger
    • 1
  • Viktor Kuncak
    • 1
  1. 1.École Polytechnique Fédérale de Lausanne (EPFL)Switzerland

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