The correlation between parity and quadratic polynomials mod 3

We prove exponentially small upper bounds on the correlation between parity and quadratic polynomials mod 3. One corollary of this is that in order to compute parity, circuits consisting of a threshold gate at the top, mod 3 gates in the middle, and AND gates of fan-in two at the inputs must be of size 2/sup /spl Omega/(n)/. This is the first result of this type for general mod subcircuits with ANDs of fan-in greater than 1. This yields an exponential improvement over a recent result of Alon and Beigel (2001). The proof uses a novel inductive estimate of the relevant exponential sums introduced by Cai et al. (1996). The exponential sum bounds are tight.

[1]  Thomas Schwentick,et al.  The Power of the Middle Bit of a #P Function , 1995, J. Comput. Syst. Sci..

[2]  David A. Mix Barrington,et al.  Bounded-width polynomial-size branching programs recognize exactly those languages in NC1 , 1986, STOC '86.

[3]  H. Niederreiter,et al.  Finite Fields: Encyclopedia of Mathematics and Its Applications. , 1997 .

[4]  A BarringtonDavid.  Bounded-width polynomial-size branching programs recognize exactly those languages in NC1 , 1989 .

[5]  Andrew Chi-Chih Yao,et al.  ON ACC and threshold circuits , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[6]  Pavel Pudlák,et al.  Threshold circuits of bounded depth , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[7]  Noga Alon,et al.  Lower bounds for approximations by low degree polynomials over Z/sub m/ , 2001, Proceedings 16th Annual IEEE Conference on Computational Complexity.

[8]  Pavel Pudlák,et al.  On the Computational Power of Depth-2 Circuits with Threshold and Modulo Gates , 1997, Theor. Comput. Sci..

[9]  Vince Grolmusz,et al.  A weight-size trade-off for circuits with MOD m gates , 1994, STOC '94.

[10]  Frederic Green.  A complex-number Fourier technique for lower bounds on the Mod-m degree , 2000, computational complexity.

[11]  Jin-Yi Cai,et al.  With probability one, a random oracle separates PSPACE from the polynomial-time hierarchy , 1986, STOC '86.

[12]  J. Håstad.  Computational limitations of small-depth circuits , 1987 .

[13]  Roman Smolensky,et al.  Algebraic methods in the theory of lower bounds for Boolean circuit complexity , 1987, STOC.

[14]  W. Schmidt.  Equations over Finite Fields: An Elementary Approach , 1976 .

[15]  Jin-Yi Cai,et al.  With probability one, a random oracle separates PSPACE from the polynomial-time hierarchy , 1986, STOC '86.

[16]  Michael Sipser,et al.  Parity, circuits, and the polynomial-time hierarchy , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[17]  Seinosuke Toda,et al.  PP is as Hard as the Polynomial-Time Hierarchy , 1991, SIAM J. Comput..

[18]  Mikael Goldmann.  A Note on the Power of Majority Gates and Modular Gates , 1995, Inf. Process. Lett..

[19]  A. Yao.  Separating the polynomial-time hierarchy by oracles , 1985 .

[20]  Noam Nisan,et al.  Multiparty Protocols, Pseudorandom Generators for Logspace, and Time-Space Trade-Offs , 1992, J. Comput. Syst. Sci..

[21]  Howard Straubing,et al.  Complex Polynomials and Circuit Lower Bounds for Modular Counting , 1992, LATIN.

[22]  Andrew Chi-Chih Yao,et al.  Separating the Polynomial-Time Hierarchy by Oracles (Preliminary Version) , 1985, FOCS.

[23]  K. Conrad,et al.  Finite Fields , 2018, Series and Products in the Development of Mathematics.

[24]  Alexander A. Razborov,et al.  Natural Proofs , 2007 .

[25]  Rudolf Lide,et al.  Finite fields , 1983 .

[26]  Eric Allender,et al.  A note on the power of threshold circuits , 1989, 30th Annual Symposium on Foundations of Computer Science.

[27]  A. Razborov.  Lower bounds on the size of bounded depth circuits over a complete basis with logical addition , 1987 .

[28]  Frederic Green,et al.  Exponential Sums and Circuits with a Single Threshold Gate and Mod-Gates , 1999, Theory of Computing Systems.

[29]  Pavel Pudlák,et al.  On the computational power of depth 2 circuits with threshold and modulo gates , 1994, STOC '94.