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TR21-105 | 22nd July 2021 11:32
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#### Functional lower bounds for restricted arithmetic circuits of depth four

**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 $\Sigma\mathord{\wedge}\Sigma\Pi$ 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 $\Sigma\mathord{\wedge}\Sigma\Pi$ 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 $\Sigma\mathord{\wedge}\Sigma\Pi$ circuits.

In this paper, for large integers $n$ and $d$ such that $\Omega(\log^2{n})\leq d\leq n^{0.01}$, we show that any $\Sigma\mathord{\wedge}\Sigma\Pi$ circuit of bounded individual degree at most $O(\frac{d}{k^2})$ 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 $GapL$.

For the sake of completeness, we also show a syntactic size lower bound against any $\Sigma\mathord{\wedge}\Sigma\Pi$ circuit computing $IMM_{n,d}$ (for the same regime of $d$) which is tight over large fields. Like Forbes, Kumar and Saptharishi [CCC, 2016], we too prove lower bounds against circuits of bounded formal degree which functionally compute $IMM_{n,d}$, for a slightly larger range of individual degree.