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• W3186058991 abstract "Recently, Forbes, Kumar and Saptharishi [CCC, 2016] proved that there exists an explicit $d^{O(1)}$-variate and degree $d$ polynomial $P_{d}in VNP$ 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 $2^{Omega(sqrt{d}log{d})}$. 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 $Sigmamathord{wedge}SigmaPi$ circuits (i.e., circuits of the form -- sums of powers of polynomials) of $2^{log^{O(1)}n}$ size. Thus they argued that a $2^{omega(log^{O(1)}{n})}$ functional lower bound for an explicit polynomial $Q$ against $Sigmamathord{wedge}SigmaPi$ 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 $Sigmamathord{wedge}SigmaPi$ circuits. In this paper, for large integers $n$ and $d$ such that $omega(log^2n)leq dleq n^{0.01}$, we show that any $Sigmamathord{wedge}SigmaPi$ circuit of bounded individual degree at most $Oleft(frac{d}{k^2}right)$ that functionally computes Iterated Matrix Multiplication polynomial $IMM_{n,d}$ ($in VP$) over ${0,1}^{n^2d}$ must have size $n^{Omega(k)}$. 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 $GapL$." @default.
• W3186058991 created "2021-08-02" @default.
• W3186058991 creator A5014415631 @default.
• W3186058991 date "2021-07-20" @default.
• W3186058991 modified "2023-09-27" @default.
• W3186058991 title "Functional lower bounds for restricted arithmetic circuits of depth four" @default.
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