Limitations of Restricted Branching in Clause Learning

  • Matti Järvisalo
  • Tommi Junttila
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4741)


The techniques for making decisions, i.e., branching, play a central role in complete methods for solving structured CSP instances. In practice, there are cases when SAT solvers benefit from limiting the set of variables the solver is allowed to branch on to so called input variables. Theoretically, however, restricting branching to input variables implies a super-polynomial increase in the length of the optimal proofs for DPLL (without clause learning), and thus input-restricted DPLL cannot polynomially simulate DPLL. In this paper we settle the case of DPLL with clause learning. Surprisingly, even with unlimited restarts, input-restricted clause learning DPLL cannot simulate DPLL (even without clause learning). The opposite also holds, and hence DPLL and input-restricted clause learning DPLL are polynomially incomparable. Additionally, we analyse the effect of input-restricted branching on clause learning solvers in practice with various structural real-world benchmarks.


Proof System Linear Temporal Logic Conjunctive Normal Form Truth Assignment Boolean Circuit 
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.
    Davis, M., Putnam, H.: A computing procedure for quantification theory. JACM 7(3), 201–215 (1960)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Davis, M., Logemann, G., Loveland, D.: A machine program for theorem proving. CACM 5(7), 394–397 (1962)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Gomes, C.P., Selman, B., Kautz, H.A.: Boosting combinatorial search through randomization. In: AAAI, pp. 431–437. AAAI Press, Stanford, California, USA (1998)Google Scholar
  4. 4.
    Marques-Silva, J.P., Sakallah, K.A.: GRASP: A search algorithm for propositional satisfiability. IEEE Trans. Comp. 48(5), 506–521 (1999)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Biere, A., Cimatti, A., Clarke, E.M., Fujita, M., Zhu, Y.: Symbolic model checking using SAT procedures instead of BDDs. In: DAC, pp. 317–320. ACM Press, New York (1999)CrossRefGoogle Scholar
  6. 6.
    Kautz, H.A., Selman, B.: Planning as satisfiability. In: ECAI, pp. 359–363. Wiley, Chichester (1992)Google Scholar
  7. 7.
    Copty, F., Fix, L., Fraer, R., Giunchiglia, E., Kamhi, G., Tacchella, A., Vardi, M.Y.: Benefits of bounded model checking at an industrial setting. In: Berry, G., Comon, H., Finkel, A. (eds.) CAV 2001. LNCS, vol. 2102, pp. 436–453. Springer, Heidelberg (2001)Google Scholar
  8. 8.
    Giunchiglia, E., Massarotto, A., Sebastiani, R.: Act, and the rest will follow: Exploiting determinism in planning as satisfiability. In: AAAI, pp. 948–953. AAAI Press, Stanford, California, USA (1998)Google Scholar
  9. 9.
    Strichman, O.: Tuning SAT checkers for bounded model checking. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, Springer, Heidelberg (2000)Google Scholar
  10. 10.
    Giunchiglia, E., Maratea, M., Tacchella, A.: Dependent and independent variables in propositional satisfiability. In: Flesca, S., Greco, S., Leone, N., Ianni, G. (eds.) JELIA 2002. LNCS (LNAI), vol. 2424, pp. 296–307. Springer, Heidelberg (2002)Google Scholar
  11. 11.
    Cook, S.A., Reckhow, R.: On the relative efficiency of propositional proof systems. J. Symb. Logic 44, 36–50 (1977)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Beame, P., Kautz, H.A., Sabharwal, A.: Towards understanding and harnessing the potential of clause learning. JAIR 22, 319–351 (2004)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Järvisalo, M., Junttila, T., Niemelä, I.: Unrestricted vs restricted cut in a tableau method for Boolean circuits. AMAI 44(4), 373–399 (2005)zbMATHGoogle Scholar
  14. 14.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1995)Google Scholar
  15. 15.
    Haken, A.: The intractability of resolution. TCS 39(2–3), 297–308 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Goerdt, A.: Regular resolution versus unrestricted resolution. SIAM J. Comp. 22(4), 661–683 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Urquhart, A.: The complexity of propositional proofs. B. Symb. Logic 1(4), 425–467 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Zhang, L., Madigan, C.F., Moskewicz, M.W., Malik, S.: Efficient conflict driven learning in boolean satisfiability solver. In: ICCAD, pp. 279–285 (2001)Google Scholar
  19. 19.
    Cook, S.A.: A short proof of the pigeon hole principle using extended resolution. SIGACT News 8(4), 28–32 (1976)CrossRefGoogle Scholar
  20. 20.
    Järvisalo, M.: Impact of restricted branching on clause learning SAT solving. Research Report A107, Helsinki University of Technology, Laboratory for Theoretical Computer Science (2007), See
  21. 21.
    Velev, M., Bryant, R.: Superscalar processor verification using efficient reductions of the logic of equality with uninterpreted functions to propositional logic. In: Pierre, L., Kropf, T. (eds.) CHARME 1999. LNCS, vol. 1703, pp. 37–53. Springer, Heidelberg (1999)Google Scholar
  22. 22.
    Pyhälä, T.: Factoring benchmarks for SAT-solvers (2004),
  23. 23.
    Järvisalo, M.: Equivalence checking multiplier designs, SAT Competition 2007 benchmark description (2007),
  24. 24.
    Jussila, T., Heljanko, K., Niemelä, I.: BMC via on-the-fly determinization. International Journal on Software Tools for Technology Transfer 7(2), 89–101 (2005)CrossRefGoogle Scholar
  25. 25.
    Latvala, T., Biere, A., Heljanko, K., Junttila, T.A.: Simple bounded LTL model checking. In: Hu, A.J., Martin, A.K. (eds.) FMCAD 2004. LNCS, vol. 3312, pp. 186–200. Springer, Heidelberg (2004)Google Scholar
  26. 26.
    Eén, N., Sörensson, N.: An extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 502–518. Springer, Heidelberg (2004)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Matti Järvisalo
    • 1
  • Tommi Junttila
    • 1
  1. 1.Helsinki University of Technology (TKK), Laboratory for Theoretical Computer Science, P.O. Box 5400, FI-02015 TKKFinland

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