Towards an Efficient Tableau Method for Boolean Circuit Satisfiability Checking

  • Tommi A. Junttila
  • Ilkka Niemelä
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1861)


Boolean circuits offer a natural, structured, and compact representation of Boolean functions for many application domains. In this paper a tableau method for solving satisfiability problems for Boolean circuits is devised. The method employs a direct cut rule combined with deterministic deduction rules. Simplification rules for circuits and a search heuristic attempting to minimize the search space are developed. Experiments in symbolic model checking domain indicate that the method is competitive against state-of-the-art satisfiability checking techniques and a promising basis for further work.


Boolean Function Conjunctive Normal Form Symbolic Model Check Boolean Circuit Input Gate 
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  1. 1.
    A. Biere, A. Cimatti, E. Clarke, and Y. Zhu. Symbolic model checking without BDDs. In W. R. Cleaveland, editor, Tools and Algorithms for the Construction and Analysis of Systems (TACAS’99), volume 1579 of LNCS, pages 193–207. Springer, 1999.CrossRefGoogle Scholar
  2. 2.
    A. Biere, A. Cimatti, E. M. Clarke, M. Fujita, and Y. Zhu. Symbolic model checking using SAT procedures instead of BDDs. In Proceedings of the 36th ACM/IEEE Design Automation Conference (DAC’99), pages 317–320. ACM, 1999.Google Scholar
  3. 3.
    A. Biere, E. Clarke, R. Raimi, and Y. Zhu. Verifying safety properties of a PowerPC microprocessor using symbolic model checking without BDDs. In N. Halbwachs and D. Peled, editors, Computer Aided Verification: 11th International Conference (CAV’99), volume 1633 of LNCS, pages 60–71. Springer, 1999.Google Scholar
  4. 4.
    A. Borälv. The industrial success of verification tools based on Stålmarck’s method. In Proceeding of the 9th International Conference on Computer Aided Verification (CAV’97), volume 1254 of LNCS, pages 7–10, Haifa, Israel, June 1997. Springer.Google Scholar
  5. 5.
    R. Bryant. Graph-based algorithms for boolean function manipulation. IEEE Transactions on Computers, 35(8):677–691, 1986.zbMATHCrossRefGoogle Scholar
  6. 6.
    J. Burch, E. Clarke, K. McMillan, D. Dill, and L. Hwang. Symbolic model checking: 1020 states and beyond. Information and Computation, 98(2):142–170, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    M. D’ Agostino and M. Mondadori. The taming of the cut. Journal of Logic and Computation, 4:285–319, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    L. Guerra e Silva, L. M. Silveira, and J. Marques-Silva. Algorithms for solving Boolean satisfiability in combinatorial circuits. In Design, Automation and Test in Europe (DATE’99), pages 526–530. IEEE, 1999.Google Scholar
  9. 9.
    T. Junttila. BCSat — a satisfiability checker for Boolean circuits. Available at
  10. 10.
    H. Kautz, D. McAllester, and B. Selman. Exploiting variable dependency in local search. A draft available at, 1997.
  11. 11.
    H. Kautz and B. Selman. Pushing the envelope: Planning, propositional logic, and stochastic search. In Proceedings of the 13th National Conference on Artificial Intelligence, Portland, Oregon, July 1996.Google Scholar
  12. 12.
    C. Li and Anbulagan. Look-ahead versus look-back for satisfiability problems. In Principles and Practice of Constraint Programming-CP97, volume 1330 of LNCS, pages 341–355. Springer, 1997.CrossRefGoogle Scholar
  13. 13.
    F. Massacci. Simplification — a general constraint propagation technique for propositional and modal tableaux. In H. de Swart, editor, Proceedings of the International Conference on Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX-98), pages 217–231. Springer, May 1998.Google Scholar
  14. 14.
    A. Nerode and R. A. Shore. Logic for Applications. Text and Monographs in Computer Science. Springer-Verlag, 1993.Google Scholar
  15. 15.
    I. Niemelä and P. Simons. Efficient implementation of the well-founded and stable model semantics. In M. Maher, editor, Proceedings of the Joint International Conference and Symposium on Logic Programming, pages 289–303. The MIT Press, 1996.Google Scholar
  16. 16.
    C. H. Papadimitriou. Computational Complexity. Addison-Wesley, 1995.Google Scholar
  17. 17.
    R. Sebastiani. Applying GSAT to non-clausal formulas. Journal of Artificial Intelligence Research, 1:309–314, 1994.zbMATHGoogle Scholar
  18. 18.
    P. Simons. Towards constraint satisfaction through logic programs and the stable model semantics. Research report A47, Helsinki University of Technology, Helsinki, Finland, August 1997. Available at Scholar
  19. 19.
    H. Zhang. SATO: An efficient propositional prover. In Automated Deduction-CADE-14, volume 1249 of LNCS, pages 272–275. Springer, 1997.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Tommi A. Junttila
    • 1
  • Ilkka Niemelä
    • 1
  1. 1.Dept. of Computer Science and Engineering Laboratory for Theoretical Computer ScienceHelsinki University of TechnologyHutFinland

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