Collections, Cardinalities, and Relations

  • Kuat Yessenov
  • Ruzica Piskac
  • Viktor Kuncak
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5944)


Logics that involve collections (sets, multisets), and cardinality constraints are useful for reasoning about unbounded data structures and concurrent processes. To make such logics more useful in verification this paper extends them with the ability to compute direct and inverse relation and function images. We establish decidability and complexity bounds for the extended logics.


Boolean Algebra Decision Procedure Description Logic Function Symbol Cardinality Constraint 
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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  1. 1.
    Baader, F., Calvanese, D., McGuinness, D., Nardi, D., Patel-Schneider, P. (eds.): The Description Logic Handbook: Theory, Implementation and Applications. CUP (2003)Google Scholar
  2. 2.
    Banerjee, A., Naumann, D.A., Rosenberg, S.: Regional logic for local reasoning about global invariants. In: Vitek, J. (ed.) ECOOP 2008. LNCS, vol. 5142, pp. 387–411. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  3. 3.
    Barnett, M., Leino, K.R.M.: Weakest-precondition of unstructured programs. In: PASTE, pp. 82–87 (2005)Google Scholar
  4. 4.
    Beyer, D., Henzinger, T.A., Jhala, R., Majumdar, R.: The software model checker BLAST. STTT 9(5-6), 505–525 (2007)CrossRefGoogle Scholar
  5. 5.
    Cohen, E., Dahlweid, M., Hillebrand, M., Leinenbach, D., Moskal, M., Santen, T., Schulte, W., Tobies, S.: VCC: A practical system for verifying concurrent c. In: TPHOLs 2009. LNCS, vol. 5674. Springer, Heidelberg (2009)Google Scholar
  6. 6.
    Cousot, P., Cousot, R.: Systematic design of program analysis frameworks. In: POPL (1979)Google Scholar
  7. 7.
    Dewar, R.K.: Programming by refinement, as exemplified by the SETL representation sublanguage. In: ACM TOPLAS (July 1979)Google Scholar
  8. 8.
    Eisenbrand, F., Shmonin, G.: Carathéodory bounds for integer cones. Operations Research Letters 34(5), 564–568 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Feferman, S., Vaught, R.L.: The first order properties of products of algebraic systems. Fundamenta Mathematicae 47, 57–103 (1959)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Givan, R., McAllester, D., Witty, C., Kozen, D.: Tarskian set constraints. Inf. Comput. 174(2), 105–131 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gulwani, S., Lev-Ami, T., Sagiv, M.: A combination framework for tracking partition sizes. In: POPL 2009, pp. 239–251 (2009)Google Scholar
  12. 12.
    Gurevich, Y., Shelah, S.: Spectra of monadic second-order formulas with one unary function. In: LICS, pp. 291–300 (2003)Google Scholar
  13. 13.
    Kuncak, V., Lam, P., Zee, K., Rinard, M.: Modular pluggable analyses for data structure consistency. IEEE Trans. Software Engineering 32(12) (December 2006)Google Scholar
  14. 14.
    Kuncak, V., Nguyen, H.H., Rinard, M.: Deciding Boolean Algebra with Presburger Arithmetic. J. of Automated Reasoning (2006)Google Scholar
  15. 15.
    Kuncak, V., Rinard, M.: Decision procedures for set-valued fields. In: 1st International Workshop on Abstract Interpretation of Object-Oriented Languages (2005)Google Scholar
  16. 16.
    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
  17. 17.
    Lewis, H.R.: Complexity results for classes of quantificational formulas. J. Comput. Syst. Sci. 21(3), 317–353 (1980)zbMATHCrossRefGoogle Scholar
  18. 18.
    Matiyasevich, Y.V.: Enumerable sets are Diophantine. Soviet Math. Doklady 11(2), 354–357 (1970)zbMATHGoogle Scholar
  19. 19.
    Ohlbach, H.J., Koehler, J.: Modal logics, description logics and arithmetic reasoning. Artificial Intelligence 109, 1–31 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Pacholski, L., Szwast, W., Tendera, L.: Complexity results for first-order two-variable logic with counting. SIAM J. on Computing 29(4), 1083–1117 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    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)Google Scholar
  22. 22.
    Piskac, R., Kuncak, V.: Decision procedures for multisets with cardinality constraints. In: Logozzo, F., Peled, D.A., Zuck, L.D. (eds.) VMCAI 2008. LNCS, vol. 4905, pp. 218–232. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  23. 23.
    Piskac, R., Kuncak, V.: Linear arithmetic with stars. In: Gupta, A., Malik, S. (eds.) CAV 2008. LNCS, vol. 5123, pp. 268–280. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  24. 24.
    Pratt-Hartmann, I.: Complexity of the two-variable fragment with counting quantifiers. Journal of Logic, Language and Information 14(3), 369–395 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Reps, T., Sagiv, M., Yorsh, G.: Symbolic implementation of the best transformer. In: Steffen, B., Levi, G. (eds.) VMCAI 2004. LNCS, vol. 2937, pp. 252–266. Springer, Heidelberg (2004)Google Scholar
  26. 26.
    Venn, J.: On the diagrammatic and mechanical representation of propositions and reasonings. Dublin Philosophical Magazine and Journal of Science 9(59), 1–18 (1880)Google Scholar
  27. 27.
    Wies, T., Kuncak, V., Lam, P., Podelski, A., Rinard, M.: Field constraint analysis. In: Emerson, E.A., Namjoshi, K.S. (eds.) VMCAI 2006. LNCS, vol. 3855, pp. 157–173. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  28. 28.
    Wies, T., Piskac, R., Kuncak, V.: Combining theories with shared set operations. In: Ghilardi, S., Sebastiani, R. (eds.) FroCoS 2009. LNCS (LNAI), vol. 5749, pp. 366–382. Springer, Heidelberg (2009)Google Scholar
  29. 29.
    Zee, K., Kuncak, V., Rinard, M.: Full functional verification of linked data structures. In: ACM PLDI (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Kuat Yessenov
    • 1
  • Ruzica Piskac
    • 2
  • Viktor Kuncak
    • 2
  1. 1.MIT Computer Science and Artificial Intelligence LabCambridgeUSA
  2. 2.EPFL School of Computer and Communication SciencesLausanneSwitzerland

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