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arXiv 2107.09703cs.CC
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深度为4的受限算术电路的函数下界

Functional lower bounds for restricted arithmetic circuits of depth four

Suryajith Chillara

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摘要：

最近，Forbes，Kumar和Saptharishi [CCC，2016]证明了存在显式的$d^{O(1)}$ -变量和次数$d$多项式$P_{d}\in VNP$，使得如果任何有界形式次数$d$的深度4电路$C$计算有界个体次数$O(1)$的多项式，即在功能上等价于$P_d$，则$C$必须具有大小$2^{Ω(\sqrt{d}\log{d})}$。他们工作的动机来自于布尔电路复杂性。基于Yao [FOCS，1985]和Beigel和Tarui [CC，1994]对$ACC^0$电路的表征，Forbes，Kumar和Saptharishi [CCC，2016]观察到$ACC^0$中的函数也可以通过代数$Σ\mathord{\wedge}ΣΠ$电路计算（即，形式的电路--多项式的幂和）大小为$2^{\log^{O(1)}n}$。因此，他们认为，一个显式多项式$Q$对$Σ\mathord{\wedge}ΣΠ$电路的$2^{ω(\log^{O(1)}{n})}$“函数”下界将意味着一个$Q$对非均匀$ACC^0$的“相应布尔函数”的下界。在他们的工作中，他们问他们的下限是否可以扩展到$Σ\mathord{\wedge}ΣΠ$电路。对大整数$n$和$d$，使得$ω(\log^2n)\leq d\leq n^{0.01}$，证明了任意次数至多为$O\left(\frac{d}{k^2}\right)$的$Σ\mathord{\wedge}ΣΠ$电路，函数计算$\{0,1\}^{n^2d}$上的迭代矩阵乘法多项式$IMM_{n,d}$（$\in VP$），其大小必为$n^{Ω(k)}$ .由于在$\{0,1\}^{n^2d}$上的迭代矩阵乘法$IMM_{n,d}$在功能上在$GapL$中，因此前述下限的改进对于个体次数的拟多项式大值保持将意味着$ACC^0$与$GapL$的细粒度分离。

英文摘要：

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 $2^{Ω(\sqrt{d}\log{d})}$. 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 $Σ\mathord{\wedge}ΣΠ$ 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 $Σ\mathord{\wedge}ΣΠ$ 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 $Σ\mathord{\wedge}ΣΠ$ circuits. In this paper, for large integers $n$ and $d$ such that $ω(\log^2n)\leq d\leq n^{0.01}$, we show that any $Σ\mathord{\wedge}ΣΠ$ circuit of bounded individual degree at most $O\left(\frac{d}{k^2}\right)$ that functionally computes Iterated Matrix Multiplication polynomial $IMM_{n,d}$ ($\in VP$) over $\{0,1\}^{n^2d}$ must have size $n^{Ω(k)}$. 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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