Lattice-Valued Binary Decision Diagrams

  • Gilles Geeraerts
  • Gabriel Kalyon
  • Tristan Le Gall
  • Nicolas Maquet
  • Jean-Francois Raskin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6252)

Abstract

This work introduces a new data structure, called Lattice-Valued Binary Decision Diagrams (or LVBDD for short), for the compact representation and manipulation of functions of the form \(\theta : 2^{\tt P}\)\({\mathcal L}\), where P is a finite set of Boolean propositions and \({\mathcal L}\) is a finite distributive lattice. Such functions arise naturally in several verification problems. LVBDD are a natural generalisation of multi-terminal ROBDD which exploit the structure of the underlying lattice to achieve more compact representations. We introduce two canonical forms for LVBDD and present algorithms to symbolically compute their conjunction, disjunction and projection. We provide experimental evidence that this new data structure can outperform ROBDD for solving the finite-word LTL satisfiability problem.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Minato, S.: Zero-suppressed BDDs for set manipulation in combinatorial problems. In: DAC 1993. ACM, New York (1993)Google Scholar
  2. 2.
    Reif Andersen, H., Hulgaard, H.: Boolean Expression Diagrams. In: LICS. IEEE, Los Alamitos (1997)Google Scholar
  3. 3.
    Devereux, B., Chechik, M.: Edge-Shifted Decision Diagrams for Multiple-Valued Logic. In: JMVLSC. Old City Publishing (2003)Google Scholar
  4. 4.
    Cousot, P., Cousot, R.: Abstract Interpretation: A Unified Lattice Model for Static Analysis of Programs by Construction or Approximation of Fixpoints. In: POPL 1977. ACM, New York (1977)Google Scholar
  5. 5.
    Clarke, E.M., Grumberg, O., Peled, D.A.: Model Checking. MIT Press, Cambridge (2000)Google Scholar
  6. 6.
    Burch, J.R., Clarke, E.M., McMillan, K.L., Dill, D.L., Hwang, J.: Symbolic Model Checking: 1020 States and Beyond. In: LICS 1990. IEEE, Los Alamitos (1990)Google Scholar
  7. 7.
    Chechik, M., Devereux, B., Easterbrook, S., Lai, A., Petrovykh, V.: Efficient Multiple-Valued Model-Checking Using Lattice Representations. In: Larsen, K.G., Nielsen, M. (eds.) CONCUR 2001. LNCS, vol. 2154, p. 441. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  8. 8.
    Cimatti, A., Clarke, E.M., Giunchiglia, F., Roveri, M.: NuSMV: A new symbolic model verifier. In: Halbwachs, N., Peled, D.A. (eds.) CAV 1999. LNCS, vol. 1633, pp. 495–499. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  9. 9.
    Delzanno, G., Raskin, J.-.F., Van Begin, L.: Covering sharing trees: a compact data structure for parameterized verification. In: STTT, vol. 5(2-3). Springer, Heidelberg (2003)Google Scholar
  10. 10.
    Bryant, R.: Graph-based Algorithms for Boolean Function Manipulation. IEEE Trans. on Comp. C-35(8) (1986)Google Scholar
  11. 11.
    Kupferman, O., Lustig, Y.: Lattice Automata. In: Cook, B., Podelski, A. (eds.) VMCAI 2007. LNCS, vol. 4349, pp. 199–213. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  12. 12.
    Fujita, M., McGeer, P.C., Yang, J.C.Y.: Multi-terminal binary decision diagrams: An efficient data structure for matrix representation. Form. Methods Syst. Des. 10(2-3) (1997)Google Scholar
  13. 13.
    De Wulf, M., Doyen, L., Henzinger, T.A., Raskin, J.F.: Antichains: A new algorithm for checking universality of finite automata. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 17–30. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. 14.
    Doyen, L., Raskin, J.F.: Antichain Algorithms for Finite Automata. In: Esparza, J., Majumdar, R. (eds.) TACAS 2010. LNCS, vol. 6015, pp. 2–22. Springer, Heidelberg (2010)Google Scholar
  15. 15.
    Rozier, K., Vardi, M.: LTL Satisfiability Checking. In: Bošnački, D., Edelkamp, S. (eds.) SPIN 2007. LNCS, vol. 4595, pp. 149–167. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  16. 16.
    Geldenhuys, J., Hansen, H.: Larger automata and less work for LTL model checking. In: Valmari, A. (ed.) SPIN 2006. LNCS, vol. 3925, pp. 53–70. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  17. 17.
    Birkhoff, G.: Lattice Theory. Colloquim Publications. Am. Math. Soc., Providence (1999)Google Scholar
  18. 18.
    Lind-Nielsen, J.: Buddy: BDD package, http://www.itu.dk/research/buddy
  19. 19.
    NuSMV Model-checker, http://nusmv.irst.itc.it/
  20. 20.
  21. 21.
    Somenzi, F.: BDD package CUDD, http://vlsi.colorado.edu/~fabio/CUDD/
  22. 22.
    Ganty, P., Maquet, N., Raskin, J.F.: Fixpoint Guided Abstraction Refinements for Alternating Automata. In: Maneth, S. (ed.) CIAA 2009. LNCS, vol. 5642, pp. 155–164. Springer, Heidelberg (2009)Google Scholar
  23. 23.
    De Wulf, M., Doyen, L., Maquet, N., Raskin, J.F.: Antichains: Alternative Algorithms for LTL Satisfiability. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 63–77. Springer, Heidelberg (2008)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Gilles Geeraerts
    • 1
  • Gabriel Kalyon
    • 1
  • Tristan Le Gall
    • 1
  • Nicolas Maquet
    • 1
  • Jean-Francois Raskin
    • 1
  1. 1.Dépt. d’Informatique (méthodes formelles et vérification)Université Libre de BruxellesBelgium

Personalised recommendations