Improved Pseudorandom Generators for Depth 2 Circuits

  • Anindya De
  • Omid Etesami
  • Luca Trevisan
  • Madhur Tulsiani
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6302)

Abstract

We prove the existence of a poly(n,m)-time computable pseudorandom generator which “1/poly(n,m)-fools” DNFs with n variables and m terms, and has seed length O(log2nm ·loglognm). Previously, the best pseudorandom generator for depth-2 circuits had seed length O(log3nm), and was due to Bazzi (FOCS 2007).

It follows from our proof that a \(1/m^{\tilde O(\log mn)}\)-biased distribution 1/poly(nm)-fools DNFs with m terms and n variables. For inverse polynomial distinguishing probability this is nearly tight because we show that for every m,δ there is a 1/mΩ(log1/δ)-biased distribution X and a DNF φ with m terms such that φ is not δ-fooled by X.

For the case of read-once DNFs, we show that seed length O(logmn ·log1/δ) suffices, which is an improvement for large δ.

It also follows from our proof that a 1/mO(log1/δ)-biased distribution δ-fools all read-once DNF with m terms. We show that this result too is nearly tight, by constructing a \(1/m^{\tilde \Omega(\log 1/\delta)}\)-biased distribution that does not δ-fool a certain m-term read-once DNF.

Keywords

DNF pseudorandom generators small bias spaces 

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References

  1. 1.
    Ajtai, M., Wigderson, A.: Deterministic simulation of probabilistic constand-depth circuits. Advances in Computing Research - Randomness and Computation 5, 199–223 (1989); Preliminary version in Proc. of FOCS 1985Google Scholar
  2. 2.
    Alon, N., Goldreich, O., Håstad, J., Peralta, R.: Simple constructions of almost k-wise independent random variables. Random Structures and Algorithms 3(3), 289–304 (1992)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alon, N., Goldreich, O., Mansour, Y.: Almost k-wise independence versus k-wise independence. Information Processing Letters 88(3), 107–110 (2003)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bazzi, L.: Minimum Distance of Error Correcting Codes versus Encoding Complexity, Symmetry, and Pseudorandomness. PhD thesis, MIT (2003)Google Scholar
  5. 5.
    Bazzi, L.: Polylogarithmic independence can fool DNF formulas. In: Proceedings of the 48th IEEE Symposium on Foundations of Computer Science, pp. 63–73 (2007)Google Scholar
  6. 6.
    Braverman, M.: Poly-logarithmic independence fools AC0 circuits. In: Proceedings of the 24th IEEE Conference on Computational Complexity, pp. 3–8 (2009)Google Scholar
  7. 7.
    Even, G., Goldreich, O., Luby, M., Nisan, N., Velickovic, B.: Approximations of general independent distributions. In: Proceedings of the 24th ACM Symposium on Theory of Computing, pp. 10–16 (1992)Google Scholar
  8. 8.
    Håstad, J.: Almost optimal lower bounds for small depth circuits. In: Proceedings of the 18th ACM Symposium on Theory of Computing, pp. 6–20 (1986)Google Scholar
  9. 9.
    Klivans, A., Lee, H., Wan, A.: Mansour’s conjecture is true for random DNF formulas. Technical Report TR10-023, Electronic Colloquium on Computational Complexity (2010)Google Scholar
  10. 10.
    Linial, N., Mansour, Y., Nisan, N.: Constant depth circuits, fourier transform and learnability. Journal of the ACM 40(3), 607–620 (1993)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Linial, N., Nisan, N.: Approximate inclusion-exclusion. Combinatorica 10(4), 349–365 (1990)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Luby, M., Velickovic, B.: On deterministic approximation of DNF. Algorithmica 16(4/5), 415–433 (1996)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Luby, M., Velickovic, B., Wigderson, A.: Deterministic approximate counting of depth-2 circuits. In: Proceedings of the 2nd ISTCS, pp. 18–24 (1993)Google Scholar
  14. 14.
    Mak, L.: Parallelism always helps. Manuscript (1993)Google Scholar
  15. 15.
    Mansour, Y.: An o(n loglogn) learning algorithm for DNF under the uniform distribution. Journal of Computer and System Sciences 50(3), 543–550 (1995)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Naor, J., Naor, M.: Small-bias probability spaces: efficient constructions and applications. SIAM Journal on Computing 22(4), 838–856 (1993)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Nisan, N.: Pseudorandom bits for constant depth circuits. Combinatorica 12(4), 63–70 (1991)CrossRefGoogle Scholar
  18. 18.
    O’Donnell, R.: Lecture notes for analysis of boolean functions (2007), http://www.cs.cmu.edu/~odonnell/boolean-analysis
  19. 19.
    Razborov, A.: A Simple Proof of Bazzi’s Theorem. ACM Trans. Comput. Theory 1(1), 1–5 (2009)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Viola, E., Wigderson, A.: Norms, XOR lemmas, and lower bounds for polynomials and protocols. Theory of Computing 4(1), 137–168 (2008)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Anindya De
    • 1
  • Omid Etesami
    • 1
  • Luca Trevisan
    • 2
  • Madhur Tulsiani
    • 3
  1. 1.University of California at Berkeley 
  2. 2.University of California at Berkeley and Stanford University 
  3. 3.Institute for Advanced Study, Princeton 

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