Functional lower bounds for restricted arithmetic circuits of depth four
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Recently, Forbes, Kumar and Saptharishi [CCC, 2016] proved that there exists an explicit d^O(1)-variate and degree d polynomial P_d∈ VNP such that if any depth four circuit C of bounded formal degree d which computes a polynomial of bounded individual degree O(1), that is functionally equivalent to P_d, then C must have size 2^Ω(√(d)logd). The motivation for their work comes from Boolean Circuit Complexity. Based on a characterization for ACC^0 circuits by Yao [FOCS, 1985] and Beigel and Tarui [CC, 1994], Forbes, Kumar and Saptharishi [CCC, 2016] observed that functions in ACC^0 can also be computed by algebraic Σ∧ΣΠ circuits (i.e., circuits of the form – sums of powers of polynomials) of 2^log^O(1)n size. Thus they argued that a 2^ω(log^O(1)n) "functional" lower bound for an explicit polynomial Q against Σ∧ΣΠ circuits would imply a lower bound for the "corresponding Boolean function" of Q against non-uniform ACC^0. In their work, they ask if their lower bound be extended to Σ∧ΣΠ circuits. In this paper, for large integers n and d such that ω(log^2n)≤ d≤ n^0.01, we show that any Σ∧ΣΠ circuit of bounded individual degree at most O(d/k^2) that functionally computes Iterated Matrix Multiplication polynomial IMM_n,d (∈ VP) over {0,1}^n^2d must have size n^Ω(k). Since Iterated Matrix Multiplication IMM_n,d over {0,1}^n^2d is functionally in GapL, improvement of the afore mentioned lower bound to hold for quasipolynomially large values of individual degree would imply a fine-grained separation of ACC^0 from GapL.
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