Multilinear Formulas, Maximal-Partition Discrepancy and Mixed-Sources Extractors

We study a new method for proving lower bounds for subclasses of arithmetic circuits. Roughly speaking, the lower bound is proved by bounding the correlation between the coefficients' vector of a polynomial and the coefficients' vector of any product of two polynomials with disjoint sets of variables. We prove lower bounds for several old and new subclasses of circuits.

[1]  Noam Nisan,et al.  Lower bounds on arithmetic circuits via partial derivatives , 2005, computational complexity.

[2]  J. Bourgain,et al.  MORE ON THE SUM-PRODUCT PHENOMENON IN PRIME FIELDS AND ITS APPLICATIONS , 2005 .

[3]  Leslie G. Valiant,et al.  Fast Parallel Computation of Polynomials Using Few Processors , 1983, SIAM J. Comput..

[4]  Andrew Chi-Chih Yao,et al.  Some complexity questions related to distributive computing(Preliminary Report) , 1979, STOC.

[5]  Troy Lee,et al.  Disjointness Is Hard in the Multi-party Number-on-the-Forehead Model , 2008, Computational Complexity Conference.

[6]  Mark Jerrum,et al.  Some Exact Complexity Results for Straight-Line Computations over Semirings , 1982, JACM.

[7]  Avi Wigderson,et al.  2-source dispersers for sub-polynomial entropy and Ramsey graphs beating the Frankl-Wilson construction , 2006, STOC '06.

[8]  Ran Raz,et al.  Multi-linear formulas for permanent and determinant are of super-polynomial size , 2004, STOC '04.

[9]  Guy Kindler,et al.  Simulating independence: new constructions of condensers, ramsey graphs, dispersers, and extractors , 2005, STOC '05.

[10]  Marc Snir,et al.  On the depth complexity of formulas , 1979, Mathematical systems theory.

[11]  Stasys Jukna.  On the P versus NP intersected with co-NP question in communication complexity , 2004, Inf. Process. Lett..

[12]  Ran Raz,et al.  A Lower Bound for the Size of Syntactically Multilinear Arithmetic Circuits , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[13]  Ran Raz.  On the Complexity of Matrix Product , 2003, SIAM J. Comput..

[14]  Leslie G. Valiant,et al.  Negation can be exponentially powerful , 1979, Theor. Comput. Sci..

[15]  Avi Wigderson,et al.  Extracting Randomness Using Few Independent Sources , 2006, SIAM J. Comput..

[16]  Ran Raz,et al.  Extractors with weak random seeds , 2005, STOC '05.

[17]  Oded Goldreich,et al.  Unbiased Bits from Sources of Weak Randomness and Probabilistic Communication Complexity , 1988, SIAM J. Comput..

[18]  Anup Rao,et al.  Extractors for a constant number of polynomially small min-entropy independent sources , 2006, STOC '06.

[19]  Martin Tompa,et al.  A Direct Version of Shamir and Snir's Lower Bounds on Monotone Circuit Depth , 1994, Inf. Process. Lett..

[20]  Ran Raz,et al.  On the complexity of matrix product , 2002, STOC '02.

[21]  Marc Snir.  On the Size Complexity of Monotone Formulas , 1980, ICALP.

[22]  Nancy A. Lynch,et al.  Proceedings of the tenth annual ACM symposium on Theory of computing , 1978 .

[23]  Ran Raz,et al.  Balancing Syntactically Multilinear Arithmetic Circuits , 2008, computational complexity.

[24]  Claus-Peter Schnorr,et al.  A Lower Bound on the Number of Additions in Monotone Computations , 1976, Theor. Comput. Sci..

[25]  H. Venkateswaran,et al.  A Lower Bound for Monotone Arithmetic Circuits Computing 0-1 Permanent , 1998, Theor. Comput. Sci..

[26]  Scott Aaronson,et al.  Multilinear formulas and skepticism of quantum computing , 2003, STOC '04.

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

[28]  Terence Tao,et al.  A sum-product estimate in finite fields, and applications , 2003, math/0301343.

[29]  Jean Bourgain,et al.  Estimates for the Number of Sums and Products and for Exponential Sums in Fields of Prime Order , 2006 .

[30]  Umesh V. Vazirani.  Towards a strong communication complexity theory or generating quasi-random sequences from two communicating slightly-random sources , 1985, STOC '85.

[31]  Black Jack,et al.  Volume 13 , 2004, Environmental Biology of Fishes.

[32]  Ran Raz,et al.  Separation of Multilinear Circuit and Formula Size , 2006, Theory Comput..

[33]  Eyal Kushilevitz,et al.  Communication Complexity , 1997, Adv. Comput..

[34]  Tak Wah Lam,et al.  Results on Communication Complexity Classes , 1992, J. Comput. Syst. Sci..