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On the Limits of Depth Reduction at Depth 3 Over Small Finite Fields

  • Suryajith Chillara
  • Partha Mukhopadhyay
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8635)

Abstract

In a surprising recent result, Gupta et al. [GKKS13b] have proved that over ℚ any n O(1)-variate and n-degree polynomial in VP can also be computed by a depth three ∑ ∏ ∑ circuit of size \(2^{O(\sqrt{n} \log^{3/2}n)}\) . Over fixed-size finite fields, Grigoriev and Karpinski proved that any ∑ ∏ ∑ circuit that computes the determinant (or the permanent) polynomial of a n×n matrix must be of size 2Ω(n). In this paper, for an explicit polynomial in VP (over fixed-size finite fields), we prove that any ∑ ∏ ∑ circuit computing it must be of size 2Ω(nlogn). The explicit polynomial that we consider is the iterated matrix multiplication polynomial of n generic matrices of size n×n. The importance of this result is that over fixed-size fields there is no depth reduction technique that can be used to compute all the n O(1)-variate and n-degree polynomials in VP by depth 3 circuits of size 2 o(nlogn). The result of [GK98] can only rule out such a possibility for ∑ ∏ ∑ circuits of size 2 o(n).

We also give an example of an explicit polynomial (NW n,ε (X)) in VNP (which is not known to be in VP), for which any ∑ ∏ ∑ circuit computing it (over fixed-size fields) must be of size 2Ω(nlogn). The polynomial we consider is constructed from the combinatorial design of Nisan and Wigderson [NW94], and is closely related to the polynomials considered in many recent papers where strong depth 4 circuit size lower bounds were shown [KSS13,KLSS14,KS13b,KS14].

Keywords

Derivative Space Univariate Polynomial Arithmetic Circuit Multilinear Polynomial Depth Reduction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Suryajith Chillara
    • 1
  • Partha Mukhopadhyay
    • 1
  1. 1.Chennai Mathematical InstituteChennaiIndia

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