# Polylogarithmic Independence Can Fool DNF Formulas

## Abstract

*k*-wise independent probability distribution on $\{0,1\}^n$ $O(m^{2.2}$ $2^{-\sqrt{k}/10})$-fools any boolean function computable by an

*m*-clause disjunctive normal form (DNF) (or conjunctive normal form (CNF)) formula on

*n*variables. Thus, for each constant $e>0$, there is a constant $c>0$ such that any boolean function computable by an

*m*-clause DNF (or CNF) formula is $m^{-e}$-fooled by any $c\log^2m$-wise probability distribution. This resolves up to an $O(\log m)$ factor the depth-2 circuit case of a conjecture due to Linial and Nisan [

*Combinatorica*, 10 (1990), pp. 349–365]. The result is equivalent to a new characterization of DNF (or CNF) formulas by low degree polynomials. It implies a similar statement for probability distributions with the small bias property. Using known explicit constructions of small probability spaces having the limited independence property or the small bias property, we directly obtain a large class of explicit pseudorandom generators of $O(\log^2m\log n)$-seed length for

*m*-clause DNF (or CNF) formulas on

*n*variables, improving previously known seed lengths.

### MSC codes

### Keywords

## Get full access to this article

View all available purchase options and get full access to this article.

## References

*Combinatorica*, 14 (1994), pp. 135–148.

*Random Structures Algorithms*, 3 (1992), pp. 289–304.

*Inform. Process. Lett.*, 88 (2003), pp. 107–110.

*Deterministic simulation of probabilistic constant depth circuits*, in Proceedings of the 26th Annual IEEE Symposium on Foundations of Computer Science, 1985, pp. 11–19.

*Minimum Distance of Error Correcting Codes versus Encoding Complexity, Symmetry, and Pseudorandomness*, Ph.D. dissertation, MIT, Cambridge, MA, 2003.

*Polylogarithmic independence can fool DNF formulas*, in Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, 2007, pp. 63–73.

*The perceptron strikes back*, in Proceedings of the 6th Annual IEEE Conference on Structure in Complexity Theory, 1991, pp. 286–291.

*SIAM J. Comput.*, 13 (1984), pp. 850–864.

*Computational Limitations for Small Depth Circuits*, Ph.D. dissertation, MIT, Cambridge, MA, 1986.

*P $=$ BPP if E requires exponential circuits: Derandomizing the XOR lemma*, in Proceedings of the 29th Annual ACM Symposium on Theory of Computing, 1997, pp. 220–229.

*The influence of variables on Boolean functions*, in Proceedings of the 29th Annual IEEE Symposium on Foundations of Computer Science, 1988, pp. 68–80.

*Harmonic analysis of switching functions*, in Recent Development in Switching Theory, Academic Press, New York, 1971, pp. 121–228.

*J. ACM*, 40 (1993), pp. 607–620.

*Combinatorica*, 10 (1990), pp. 349–365.

*A simple parallel algorithm for the maximal independent set problem*, in Proceedings of the 17th Annual ACM Symposium on Theory of Computing, 1985, pp. 1–10.

*Algorithmica*, 16 (1996), pp. 415–433.

*Deterministic approximate counting of depth-2 circuits*, in Proceedings of the 2nd Israel Symposium on the Theory and Computing Systems, 1993, pp. 18–24.

*SIAM J. Comput.*, 22 (1993), pp. 838–856.

*Combinatorica*, 12 (1991), pp. 63–70.

*Comput. Complexity*, 4 (1994), pp. 301–313.

*Hardness vs. randomness*, in Proceedings of the 29th Annual IEEE Symposium on Foundations of Computer Science, 1988, pp. 2–11.

*Math. Zametki*, 41 (1987), pp. 598–607.

*A Simple Proof of Bazzi's Theorem*, Report TR08-081, Electronic Colloquium on Computational Complexity, 2008.

*Enumerative Combinatorics*, Vol. I, Cambridge University Press, Cambridge, UK, 1997.

*A note on deterministic approximate counting for k-DNF*, in Proceedings of the APPROX-RANDOM, 2004, pp. 417–426.

*Randomness, Adversaries, and Computation*, Ph.D. dissertation, University of California, Berkeley, CA, 1986.

*Theory and applications of trapdoor functions*, in Proceedings of the 23rd IEEE Annual Symposium on Foundations of Computer Science, 1982, pp. 80–91.

## Information & Authors

### Information

#### Published In

#### Copyright

#### History

**Submitted**: 18 May 2007

**Accepted**: 24 September 2008

**Published online**: 4 March 2009

#### MSC codes

#### Keywords

### Authors

## Metrics & Citations

### Metrics

### Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

#### Cited By

- Beyond Natural Proofs: Hardness Magnification and LocalityJournal of the ACM, Vol. 69, No. 4 | 31 Aug 2022
- Fooling PolytopesJournal of the ACM, Vol. 69, No. 2 | 31 January 2022
- An improved derandomization of the switching lemmaProceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing | 15 June 2021
- Bounded Independence versus Symmetric TestsACM Transactions on Computation Theory, Vol. 11, No. 4 | 11 July 2019
- Fooling polytopesProceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing | 23 June 2019
- Non-Malleable Codes for Small-Depth Circuits2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS) | 1 Oct 2018
- Bounded Independence Plus Noise Fools ProductsSIAM Journal on Computing, Vol. 47, No. 2 | 12 April 2018
- An Average-Case Depth Hierarchy Theorem for Boolean CircuitsJournal of the ACM, Vol. 64, No. 5 | 29 August 2017
- Pseudorandom Generators with Optimal Seed Length for Non-Boolean Poly-Size CircuitsACM Transactions on Computation Theory, Vol. 9, No. 2 | 27 April 2017
- Small-Bias is Not Enough to Hit Read-Once CNFTheory of Computing Systems, Vol. 60, No. 2 | 7 May 2016
- Bounded Indistinguishability and the Complexity of Recovering SecretsAdvances in Cryptology – CRYPTO 2016 | 21 July 2016
- Substitution-Permutation Networks, Pseudorandom Functions, and Natural ProofsJournal of the ACM, Vol. 62, No. 6 | 10 December 2015
- An Average-Case Depth Hierarchy Theorem for Boolean Circuits2015 IEEE 56th Annual Symposium on Foundations of Computer Science | 1 Oct 2015
- Mining Circuit Lower Bound Proofs for Meta-Algorithmscomputational complexity, Vol. 24, No. 2 | 21 April 2015
- Pseudorandom generators for combinatorial checkerboardscomputational complexity, Vol. 22, No. 4 | 4 February 2012
- Pseudorandom generators for CC0[p] and the Fourier spectrum of low-degree polynomials over finite fieldscomputational complexity, Vol. 22, No. 4 | 23 November 2012
- DNF sparsification and a faster deterministic counting algorithmcomputational complexity, Vol. 22, No. 2 | 7 May 2013
- Pseudorandom Generators for Combinatorial ShapesSIAM Journal on Computing, Vol. 42, No. 3 | 30 May 2013
- Better Pseudorandom Generators from Milder Pseudorandom Restrictions2012 IEEE 53rd Annual Symposium on Foundations of Computer Science | 1 Oct 2012
- Bounded-Depth Circuits Cannot Sample Good Codescomputational complexity, Vol. 21, No. 2 | 30 March 2012
- DNF Sparsification and a Faster Deterministic Counting Algorithm2012 IEEE 27th Conference on Computational Complexity | 1 Jun 2012
- Substitution-Permutation Networks, Pseudorandom Functions, and Natural ProofsAdvances in Cryptology – CRYPTO 2012 | 1 Jan 2012
- Patterns hidden from simple algorithmsCommunications of the ACM, Vol. 54, No. 4 | 1 April 2011
- Poly-logarithmic independence fools bounded-depth boolean circuitsCommunications of the ACM, Vol. 54, No. 4 | 1 April 2011
- Fooling Functions of Halfspaces under Product Distributions2010 IEEE 25th Annual Conference on Computational Complexity | 1 Jun 2010
- Polylogarithmic independence fools AC0 circuitsJournal of the ACM, Vol. 57, No. 5 | 25 June 2008
- A Simple Proof of Bazzi’s TheoremACM Transactions on Computation Theory, Vol. 1, No. 1 | 1 February 2009

## View Options

### Get Access

**Access via your Institution**- Questions about how to access this content? Contact SIAM at
**[email protected]**.