Lower Bounds for Splittings by Linear Combinations

  • Dmitry Itsykson
  • Dmitry Sokolov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8635)

Abstract

A typical DPLL algorithm for the Boolean satisfiability problem splits the input problem into two by assigning the two possible values to a variable; then it simplifies the two resulting formulas. In this paper we consider an extension of the DPLL paradigm. Our algorithms can split by an arbitrary linear combination of variables modulo two. These algorithms quickly solve formulas that explicitly encode linear systems modulo two, which were used for proving exponential lower bounds for conventional DPLL algorithms.

We prove exponential lower bounds on the running time of DPLL with splitting by linear combinations on 2-fold Tseitin formulas and on formulas that encode the pigeonhole principle.

Raz and Tzameret introduced a system R(lin) which operates with disjunctions of linear equalities with integer coefficients. We consider an extension of the resolution proof system that operates with disjunctions of linear equalities over \(\mathbb{F}_2\); we call this system Res-Lin. Res-Lin can be p-simulated in R(lin) but currently we do not know any superpolynomial lower bounds in R(lin). Tree-like proofs in Res-Lin are equivalent to the behavior of our algorithms on unsatisfiable instances. We prove that Res-Lin is implication complete and also prove that Res-Lin is polynomially equivalent to its semantic version.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alekhnovich, M., Hirsch, E.A., Itsykson, D.: Exponential lower bounds for the running time of DPLL algorithms on satisfiable formulas. J. Autom. Reason. 35(1-3), 51–72 (2005)MATHMathSciNetGoogle Scholar
  2. 2.
    Beame, P., Pitassi, T., Segerlind, N.: Lower bounds for lovász-schrijver systems and beyond follow from multiparty communication complexity. SIAM Journal on Computing 37(3), 845–869 (2007)MATHMathSciNetGoogle Scholar
  3. 3.
    Ben-Sasson, E., Wigderson, A.: Short proofs are narrow — resolution made simple. Journal of ACM 48(2), 149–169 (2001)MATHMathSciNetGoogle Scholar
  4. 4.
    Cook, S.A., Reckhow, R.A.: The relative efficiency of propositional proof systems. The Journal of Symbolic Logic 44(1), 36–50 (1979)MATHMathSciNetGoogle Scholar
  5. 5.
    Davis, M., Logemann, G., Loveland, D.: A machine program for theorem-proving. Communications of the ACM 5, 394–397 (1962)MATHMathSciNetGoogle Scholar
  6. 6.
    Davis, M., Putnam, H.: A computing procedure for quantification theory. Journal of the ACM 7, 201–215 (1960)MATHMathSciNetGoogle Scholar
  7. 7.
    Demenkov, E., Kulikov, A.S.: An elementary proof of a 3no(n) lower bound on the circuit complexity of affine dispersers. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 256–265. Springer, Heidelberg (2011)Google Scholar
  8. 8.
    Itsykson, D., Sokolov, D.: The complexity of inversion of explicit goldreich’s function by DPLL algorithms. In: Kulikov, A., Vereshchagin, N. (eds.) CSR 2011. LNCS, vol. 6651, pp. 134–147. Springer, Heidelberg (2011)Google Scholar
  9. 9.
    Itsykson, D.: Lower bound on average-case complexity of inversion of goldreich’s function by drunken backtracking algorithms. Theory Comput. Syst. 54(2), 261–276 (2014)MathSciNetGoogle Scholar
  10. 10.
    Kalyanasundaram, B., Schintger, G.: The probabilistic communication complexity of set intersection. SIAM J. Discret. Math. 5(4), 545–557 (1992)MATHGoogle Scholar
  11. 11.
    Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, New York (1997)MATHGoogle Scholar
  12. 12.
    Raz, R., Tzameret, I.: Resolution over linear equations and multilinear proofs. Ann. Pure Appl. Logic 155(3), 194–224 (2008)MATHMathSciNetGoogle Scholar
  13. 13.
    Razborov, A.A.: Pseudorandom generators hard for k-dnf resolution and polynomial calculus resolution. Technical report (2003)Google Scholar
  14. 14.
    Seto, K., Tamaki, S.: A satisfiability algorithm and average-case hardness for formulas over the full binary basis. Computational Complexity 22(2), 245–274 (2013)MATHMathSciNetGoogle Scholar
  15. 15.
    Tseitin, G.S.: On the complexity of derivation in the propositional calculus. Zapiski Nauchnykh Seminarov LOMI 8, 234–259 (1968); English translation of this volume: Consultants Bureau, N.Y., pp. 115–125 (1970)Google Scholar
  16. 16.
    Urquhart, A.: The depth of resolution proofs. Studia Logica 99(1-3), 249–364 (2011)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Dmitry Itsykson
    • 1
  • Dmitry Sokolov
    • 1
  1. 1.Steklov Institute of Mathematics at St.PetersburgSt.PetersburgRussia

Personalised recommendations