On DNF Approximators for Monotone Boolean Functions

  • Eric Blais
  • Johan Håstad
  • Rocco A. Servedio
  • Li-Yang Tan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8572)

Abstract

We study the complexity of approximating monotone Boolean functions with disjunctive normal form (DNF) formulas, exploring two main directions. First, we construct DNF approximators for arbitrary monotone functions achieving one-sided error: we show that every monotone f can be ε-approximated by a DNF g of size \(2^{n-\Omega_\epsilon(\sqrt{n})}\) satisfying g(x) ≤ f(x) for all x ∈ {0,1}n. This is the first non-trivial universal upper bound even for DNF approximators incurring two-sided error.

Next, we study the power of negations in DNF approximators for monotone functions. We exhibit monotone functions for which non-monotone DNFs perform better than monotone ones, giving separations with respect to both DNF size and width. Our results, when taken together with a classical theorem of Quine [1], highlight an interesting contrast between approximation and exact computation in the DNF complexity of monotone functions, and they add to a line of work on the surprising role of negations in monotone complexity [2,3,4].

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References

  1. 1.
    Quine, W.V.O.: Two theorems about truth functions. Bol. Soc. Math. Mexicana 10, 64–70 (1954)MathSciNetGoogle Scholar
  2. 2.
    Razborov, A.A.: Lower bounds for the monotone complexity of some boolean functions. Soviet Mathematics Doklady 31, 354–357 (1985)MATHGoogle Scholar
  3. 3.
    Okol’nishnikova, E.: On the influence of negations on the complexity of a realization of monotone Boolean functions by formulas of bounded depth. Metody Diskret. Analiz. 38, 74–80 (1982) (in Russian)Google Scholar
  4. 4.
    Ajtai, M., Grevich, Y.: Monotone versus positive. Journal of the ACM 34(4), 1004–1015 (1987)CrossRefMATHGoogle Scholar
  5. 5.
    Korshunov, A.D.: Monotone Boolean functions. Russian Math. Surveys (Uspekhi Mat. Nauk) 58(5), 929–1001 (2003)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Alon, N., Boppana, R.: The monotone circuit complexity of Boolean functions. Combinatorica 7, 1–22 (1987)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Karchmer, M., Wigderson, A.: Monotone circuits for connectivity require super-logarithmic depth. In: STOC 1988: Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, pp. 539–550. ACM, New York (1988)CrossRefGoogle Scholar
  8. 8.
    Tardos, É.: The gap between monotone and non-monotone circuit complexity is exponential. Combinatorica 8(1), 141–142 (1988)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Raz, R., Wigderson, A.: Monotone circuits for matching require linear depth. In: Proceedings of the 22nd ACM Symposium on Theory of Computing, pp. 287–292 (1990)Google Scholar
  10. 10.
    Karchmer, M., Raz, R., Wigderson, A.: Super-logarithmic depth lower bounds via direct sum in communication coplexity. In: Structure in Complexity Theory Conference, pp. 299–304 (1991)Google Scholar
  11. 11.
    Grigni, M., Sipser, M.: Monotone separation of logarithmic space from logarithmic depth. J. Comput. Syst. Sci. 50(3), 433–437 (1995)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Razborov, A., Rudich, S.: Natural proofs. Journal of Computer and System Sciences 55(1), 24–35 (1997)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Goldmann, M., Håstad, J.: Monotone circuits for connectivity have depth (log n)2 − o(1). SIAM J. Comput. 27(5), 1283–1294 (1998)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Raz, R., McKenzie, P.: Separation of the monotone NC hierarchy. Combinatorica 19(3), 403–435 (1999)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Potechin, A.: Bounds on monotone switching networks for directed connectivity. In: Symposium on Foundations of Computer Science (FOCS), pp. 553–562 (2010)Google Scholar
  16. 16.
    Chan, S.M., Potechin, A.: Tight bounds for monotone switching networks via fourier analysis. In: Symposium on Theory of Computing (STOC), pp. 495–504 (2012)Google Scholar
  17. 17.
    Filmus, Y., Pitassi, T., Robere, R., Cook, S.A.: Average case lower bounds for monotone switching networks. In: Symposium on Foundations of Computer Science (FOCS) (2013)Google Scholar
  18. 18.
    O’Donnell, R.T., Wimmer, K.: Approximation by DNF: examples and counterexamples. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 195–206. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  19. 19.
    Blais, E., Tan, L.Y.: Approximating Boolean functions with depth-2 circuits. In: Proceedings of the 28th Annual IEEE Conference on Computational Complexity, pp. 74–85 (2013)Google Scholar
  20. 20.
    Bshouty, N., Tamon, C.: On the Fourier spectrum of monotone functions. Journal of the ACM 43(4), 747–770 (1996)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Jackson, J.: An efficient membership-query algorithm for learning DNF with respect to the uniform distribution. Journal of Computer and System Sciences 55(3), 414–440 (1997)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Krause, M., Pudlák, P.: On the computational power of depth-2 circuits with threshold and modulo gates. Theoretical Computer Science 174(1–2), 137–156 (1997)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Friedgut, E.: Boolean functions with low average sensitivity depend on few coordinates. Combinatorica 18(1), 27–36 (1998)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Amano, K.: Tight bounds on the average sensitivity of k-CNF. Theory of Computing 7(1), 45–48 (2011)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Gopalan, P., Meka, R., Reingold, O.: Dnf sparsification and a faster deterministic counting algorithm. Computational Complexity 22(2), 275–310 (2013)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Kalai, G.: Noise stability and threshold circuits. Gil Kalai’s, Combinatorics and more, blog (2010), http://gilkalai.wordpress.com/2010/02/10/noise-stability-and-threshold-circuits/

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Eric Blais
    • 1
  • Johan Håstad
    • 2
  • Rocco A. Servedio
    • 3
  • Li-Yang Tan
    • 3
  1. 1.MITUSA
  2. 2.KTH Royal Institute of TechnologyUSA
  3. 3.Columbia UniversityUSA

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