Improved Average-Case Lower Bounds for De Morgan Formula Size: Matching Worst-Case Lower Bound

We give an explicit function $h:\{0,1\}^n \to \{0,1\}$ such that every de Morgan formula of size $n^{3-o(1)}/r^2$ agrees with $h$ on at most a fraction of $\frac{1}{2}+2^{-\Omega(r)}$ of the inputs. Our technical contributions include a theorem that shows that the “expected shrinkage” result of H\aastad [SIAM J. Comput., 27 (1998), pp. 48--64] actually holds with very high probability (where the restrictions are chosen from a certain distribution that takes into account the structure of the formula), using ideas of Impagliazzo, Meka, and Zuckerman [Proceedings of FOCS, 2012, pp. 111--119].

[1]  Ben Reichardt,et al.  Reflections for quantum query algorithms , 2010, SODA '11.

[2]  Johan Hå stad.  The Shrinkage Exponent of de Morgan Formulas is 2 , 1998 .

[3]  Anup Rao,et al.  An Exposition of Bourgain's 2-Source Extractor , 2007, Electron. Colloquium Comput. Complex..

[4]  H. Chernoff.  A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations , 1952 .

[5]  Ran Raz,et al.  Average-case lower bounds for formula size , 2013, STOC '13.

[6]  Ronald de Wolf,et al.  Quantum lower bounds by polynomials , 2001, JACM.

[7]  Uri Zwick,et al.  Shrinkage of de Morgan Formulae under Restriction , 1993, Random Struct. Algorithms.

[8]  Russell Impagliazzo,et al.  Pseudorandomness from Shrinkage , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[9]  David Zuckerman,et al.  Mining Circuit Lower Bound Proofs for Meta-Algorithms , 2014, computational complexity.

[10]  Valentine Kabanets,et al.  Compression of Boolean Functions , 2013, Electron. Colloquium Comput. Complex..

[11]  Ran Raz,et al.  Deterministic Extractors for Bit-Fixing Sources by Obtaining an Independent Seed , 2006, SIAM J. Comput..

[12]  V. M. Khrapchenko.  Method of determining lower bounds for the complexity of P-schemes , 1971 .

[13]  Jean Bourgain,et al.  On the Construction of Affine Extractors , 2007 .

[14]  Stasys Jukna,et al.  Boolean Function Complexity Advances and Frontiers , 2012, Bull. EATCS.

[15]  Anup Rao,et al.  Extractors for Low-Weight Affine Sources , 2009, 2009 24th Annual IEEE Conference on Computational Complexity.

[16]  Avishay Tal.  Shrinkage of De Morgan Formulae from Quantum Query Complexity , 2014, Electron. Colloquium Comput. Complex..

[17]  Oded Goldreich,et al.  The Bit Extraction Problem of t-Resilient Functions (Preliminary Version) , 1985, FOCS.

[18]  Ran Raz,et al.  Improved Average-Case Lower Bounds for DeMorgan Formula Size , 2013, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.

[19]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[20]  Rajeev Motwani,et al.  Randomized algorithms , 1996, CSUR.

[21]  Noam Nisan,et al.  The Effect of Random Restrictions on Formula Size , 1993, Random Struct. Algorithms.

[22]  David Zuckerman,et al.  DETERMINISTIC EXTRACTORS FOR BIT-FIXING SOURCES AND EXPOSURE-RESILIENT CRYPTOGRAPHY , 2003 .