On the limitations of ordered representations of functions

  • Jayram S. Thathachar
Regular Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1427)


We demonstrate the limitations of various ordered representations that have been considered in the literature for symbolic model checking including BDDs [3], +-BMDs [6], HDDs [15], MTBDDs [13] and EVBDDs [25]. We introduce a lower bound technique that applies to a broad spectrum of such functional representations. Using an abstraction that encompasses all these representations, we apply this technique to show exponential size bounds for a wide range of integer and boolean functions that arise in symbolic model checking in the definition and implicit exploration of the state spaces. We give the first examples of integer functions including integer division, remainder, high/low-order words of multiplication, square root and reciprocal that require exponential size in all these representations. Finally, we show that there is a simple regular language that requires exponential size to be represented by any +-BMD, even though BDDs can represent any regular language in linear size.


  1. 1.
    B. Becker, R. Drechsler, and R. Enders. On the computational power of bit-level and word level decision diagrams. In 4. GI/ITG/GME Workshop zur Methoden des Entwurfs und der Verifikation Digitaler Systeme, Berichte aus der Informatik, pages 71–80, Kreischa, March 1996. Shaker Verlag, Aachen.Google Scholar
  2. 2.
    Amos Beimel, Francesco Bergadano, Nader H. Bshouty, Eyal Kushilevitz, and Stefano Varricchio. On the applications of multiplicity automata in learning. In 37th Annual Symposium on Foundations of Computer Science, Burlington, Vermont, 14–16 October 1996. IEEE.Google Scholar
  3. 3.
    R. E. Bryant. Graph-based algorithms for boolean function manipulation. IEEE Transactions on Computers, C-35(8):677–691, August 1986.Google Scholar
  4. 4.
    R. E. Bryant. On the complexity of VLSI implementations and graph representations of boolean functions with application to integer multiplication. IEEE Transactions on Computers, 40(2):205–213, February 1991.CrossRefGoogle Scholar
  5. 5.
    R. E. Bryant. Binary decision diagrams and beyond: Enabling technologies for formal verification. In International Conference on Computer Aided Design, pages 236–245, Los Alamitos, Ca., USA, November 1995. IEEE Computer Society Press.Google Scholar
  6. 6.
    R.E. Bryant and Y.-A. Chen. Verification of arithmetic circuits with binary moment diagrams. In 32nd ACM/IEEE Design Automation Conference, Pittsburgh, June 1995.Google Scholar
  7. 7.
    R.E. Bryant and Y.-A. Chen. Bit-level analysis of an SRT divider circuit. In 33rd ACM/IEEE Design Automation Conference, 1996.Google Scholar
  8. 8.
    J.R. Burch, E.M. Clarke, D.E. Long, K.L. MacMillan, and D.L. Dill. Symbolic model checking for sequential circuit verification. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 13(4):401–424, April 1994.CrossRefGoogle Scholar
  9. 9.
    J.R. Burch, E.M. Clarke, K.L. McMillan, D.L. Dill, and L.J. Hwang. Symbolic model checking: 1020 states and beyond. In Proceedings of the Fifth Annual IEEE Symposium on Logic in Computer Science, pages 1–33, Washington, D.C., June 1990. IEEE CS Press.Google Scholar
  10. 10.
    J. W. Carlyle and A. Paz. Realizations by stochastic finite automata. Journal of Computer and System Sciences, 5(1):26–40, February 1971.Google Scholar
  11. 11.
    Y-A. Chen, E. Clarke, P H. Ho, Y Hoskote, T Kam, M. Khaira, J. O' Leary, and X. Zhao. Verification of all circuits in a foating-pont unit using word-level model checking. In First International Conference on Formal Methods in Computer-Aided Design, volume 1166 of Lecture Notes Comp. Sci., pages 19–33, Palo Alto, CA, November 1996. Springer Verlag.Google Scholar
  12. 12.
    Ying-An Chen and R.E. Bryant. +PHDD: an efficient graph representation for floating point circuit verification. In International Conference on Computer Aided Design, pages 2–7, Los Alamitos, Ca., USA, November 1997. IEEE Computer Society PressGoogle Scholar
  13. 13.
    E. Clarke, K.L. McMillian, X. Zhao, M. Fujita, and J.C.-Y Yang. Spectral transforms for large boolean functions with application to technologic mapping. In 30th ACM/IEEE Design Automation Conference, pages 54-60, Dallas, TX, June 1993.Google Scholar
  14. 14.
    E. M. Clarke and E. A. Emerson. Synthesis of synchronization skeletons from branching time temporal logic. Lecture Notes Comp. Sci., 131:52–71, 1982.Google Scholar
  15. 15.
    E. M. Clarke, M. Fujita, and X. Zhao. Hybrid decision diagrams-overcoming limitations of MTBDDs and BMDs. In International Conference on Computer Aided Design, pages 159–163, Los Alamitos, CA, November 1995. IEEE Computer Society Press.Google Scholar
  16. 16.
    E. M. Clarke, S. M. German, and X. Zhao. Verifying the SRT division algorithm using theorem proving techniques. Lecture Notes in Computer Science, 1102, 1996.Google Scholar
  17. 17.
    Raymund M. Clarke and Jeanette M. Wing.Formal methods: State of the art and future directions. ACM Computing Surveys, 28(4):626–643, December 1996.CrossRefGoogle Scholar
  18. 18.
    M. Diezfelbinger, J. Hromkovic, and G. Schnitger. A comparison of two lower bound methods for communication complexity. In Symposium on Mathematical Foundations of Computer Science, pages 326–335, 1994.Google Scholar
  19. 19.
    R. Enders. Note on the complexity of binary moment diagram representations. In IFIP WG 10.5 Workshop on Applications of Reed-Muller Expansion in Circuit Design, pages 191–197, 1995.Google Scholar
  20. 20.
    M. Fliess. Matrices de Hankel. J. Math. Pures et Appl., 53:197–224, 1974.Google Scholar
  21. 21.
    J. Gergov and Ch. Meinel. Efficient boolean manipulation with OBDD's can be extended to read-once only branching programs. IEEE Transactions on Computers, 43(10):1197–1209, October 1994.CrossRefGoogle Scholar
  22. 22.
    András Hajnal, Wolfgang Maass, and György Turán. On the communication complexity of graph properties. In Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, pages 186–191, Chicago, Illinois, 2–4 May 1988.Google Scholar
  23. 23.
    Harju and Karhumaki. The equivalence problem of multitape finite automata. Theoretical Computer Science, 78, 1991.Google Scholar
  24. 24.
    Eyal Kushilevitz and Noam Nisan. Communication complexity. Cambridge University Press, Cambridge [England]; New York, 1997.Google Scholar
  25. 25.
    Y.-T. Lai and S. Sastry. Edge-valued binary decision diagrams for multi-level hierarchical verification. In 29th ACM/IEEE Design Automation Conference, pages 608–613, 1992.Google Scholar
  26. 26.
    Tak Wah Lam and Larry Ruzzo. Results on communication complexity classes. Journal of Computer and System Sciences, 44, 1992.Google Scholar
  27. 27.
    Thomas Lengauer. VLSI theory. In Handbook of Theoretical Computer Science, volume 1. The MIT Press/Elsevier, 1990.Google Scholar
  28. 28.
    Richard J. Lipton and Robert Sedgewick. Lower bounds for VLSI. In Conference Proceedings of the Thirteenth Annual ACM Symposium on Theory of Computation, pages 300–307, Milwaukee, Wisconsin, 11–13 May 1981.Google Scholar
  29. 29.
    K.L. McMillan. Symbolic Model Checking. Kluwer Academic Publishers, 1993.Google Scholar
  30. 30.
    Kurt Mehlhorn and Erik M. Schmidt. Las Vegas is better than determinism in VLSI and distributed computing (extended abstract). In Proceedings of the Fourteenth Annual ACM Symposium on Theory of Computing, pages 330–337, San Francisco, California, May 1982.Google Scholar
  31. 31.
    C. Papadimitriou and M. Sipser. Communication complexity. Journal of Computer and System Sciences, 28, 1984.Google Scholar
  32. 32.
    Stephen Ponzio. A lower bound for integer multiplication with read-once branching programs. In Proceedings of the Twenty-Seventh Annual ACM Symposium on Theory of Computing, pages 130–139, Las Vegas, Nevada, 29 May-1 June 1995.Google Scholar
  33. 33.
    Andrew Chi-Chih Yao. Some complexity questions related to distributive computing (preliminary report). In Conference Record of the Eleventh Annual ACM Symposium on Theory of Computing, pages 209–213, Atlanta, Georgia, 30 April-2 May 1979.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Jayram S. Thathachar
    • 1
  1. 1.Department of Computer Science and EngineeringUniversity of WashingtonSeattle, Washington

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