On the limits of depth reduction at depth 3 over small finite fields
Information and Computation
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 ΣΠΣ circuit of size 2 O( √ n log 3/2 n) 1 . Over fixed-size finite fields, Grigoriev and Karpinski proved that any ΣΠΣ circuit that computes the determinant (or the permanent) polynomial of a n × n matrix must be of size 2 Ω(n) . In this paper, for an explicit polynomial in VP (over fixed-size finite fields), we prove that any ΣΠΣ
... ircuit computing it must be of size 2 Ω(n log n) . 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(n log n) . The result of [GK98] can only rule out such a possibility for ΣΠΣ circuits of size 2 o(n) . We also give an example of an explicit polynomial (NWn, (X)) in VNP (which is not known to be in VP), for which any ΣΠΣ circuit computing it (over fixed-size fields) must be of size 2 Ω(n log n) . 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 are shown [KSS13, KLSS14, KS13b, KS14].