SIAM Journal on Computing, Volume 50, Issue 4, Page 1461-1499, January 2021.
In this paper, we prove the first fixed-depth size-hierarchy theorem for uniform ${\mathrm{AC}}^0[\oplus]$. In particular, we show that for any fixed $d$ and integer parameter $k$, the class ${\mathcal{{C}}}_{d,k}$ of functions that have uniform ${\mathrm{AC}}^0[\oplus]$ formulas of depth $d$ and size $n^k$ form an infinite hierarchy. We show this by exhibiting the first class of functions that have uniform ${\mathrm{AC}}^0[\oplus]$ formulas of size $n^k$ but no ${\mathrm{AC}}^0[\oplus]$ formulas of size less than $n^{\varepsilon_0 k}$ for some absolute constant $\varepsilon_0 > 0$. The uniform formulas are designed to solve the $\delta$-coin problem, which is the computational problem of distinguishing between coins that are heads with probability $(1+\delta)/2$ or $(1-\delta)/2,$ where $\delta$ is a parameter that is going to $0$. We study the complexity of this problem and make progress on both upper bound and lower bound fronts. Regarding Upper bounds, for any constant $d\geq 2$, we show that there are uniform monotone ${\mathrm{AC}}^0$ formulas (i.e., made up of AND and OR gates only) solving the $\delta$-coin problem that have depth $d$, size $\exp(O(d\cdot(1/\delta)^{1/(d-1)}))$, and sample complexity (i.e., number of inputs) ${\mathop{\mathrm{poly}}}(1/\delta).$ This matches previous upper bounds of O'Donnell and Wimmer [ICALP 2007: Automata, Languages and Programming, Lecture Notes in Comput. Sci. 4596, Springer, New York, 2007, pp. 195--206] and Amano [ICALP 2009: Automata, Languages and Programming, Lecture Notes in Comput. Sci. 5555, Springer, New York, 2009, pp. 59--70] in terms of size (which is optimal), while improving the sample complexity from $\exp(O(d\cdot(1/\delta)^{1/(d-1)}))$ to ${\mathop{\mathrm{poly}}}(1/\delta)$. The improved sample complexity is crucial for proving the size-hierarchy theorem. Regarding Lower bounds, we show that the preceding upper bounds are nearly tight (in terms of size) even for the significantly stronger model of ${\mathrm{AC}}^0[\oplus]$ formulas (which are also allowed NOT and Parity gates): formally, we show that any ${\mathrm{AC}}^0[\oplus]$ formula solving the $\delta$-coin problem must have size $\exp(\Omega(d\cdot(1/\delta)^{1/(d-1)})).$ This strengthens a result of Shaltiel and Viola [SIAM J. Comput., 39 (2010), pp. 3122--3154], who prove an $\exp(\Omega((1/\delta)^{1/(d+2)}))$ lower bound for ${\mathrm{AC}}^0[\oplus]$ circuits, and a result of Cohen, Ganor, and Raz [APPROX-RANDOM, LIPIcs. Leibniz Int. Proc. Inform. 28, Schloss Dagstuhl, Leibniz-Zentrum fuer Informatik, Wadern, 2014, pp. 618--629], who show an $\exp(\Omega((1/\delta)^{1/(d-1)}))$ lower bound for ${\mathrm{AC}}^0$ circuits. The upper bound is a derandomization involving a use of Janson's inequality and an extension of classical polynomial-based combinatorial designs. For the lower bound, we prove an optimal (up to a constant factor) degree lower bound for multivariate polynomials over ${\mathbb{F}}_2$ solving the $\delta$-coin problem, which may be of independent interest.

中文翻译：

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$\mathrm{AC}^0[\oplus]$ 的固定深度大小层次定理通过硬币问题

SIAM Journal on Computing，第 50 卷，第 4 期，第 1461-1499 页，2021 年 1 月。
在本文中，我们证明了统一 ${\mathrm{AC}}^0[\oplus]$ 的第一个固定深度大小层次定理。特别地，我们证明对于任何固定的 $d$ 和整数参数 $k$，具有统一 ${\mathrm{AC} 的函数类 ${\mathcal{{C}}}_{d,k}$ }^0[\oplus]$ 深度$d$ 和大小$n^k$ 的公式形成了一个无限的层次结构。我们通过展示具有统一 ${\mathrm{AC}}^0[\oplus]$ 大小 $n^k$ 但没有 ${\mathrm{AC}}^0[\ oplus]$ 大小小于 $n^{\varepsilon_0 k}$ 的公式，用于某些绝对常数 $\varepsilon_0 > 0$。统一公式旨在解决 $\delta$-coin 问题，这是区分正面硬币的计算问题，概率为 $(1+\delta)/2$ 或 $(1-\delta)/2 ,$ 其中 $\delta$ 是指向 $0$ 的参数。我们研究了这个问题的复杂性，并在上限和下限方面取得了进展。关于上限，对于任何常数 $d\geq 2$，我们证明有统一的单调 ${\mathrm{AC}}^0$ 公式（即，仅由 AND 和 OR 门组成）解决 $\delta具有深度 $d$、大小 $\exp(O(d\cdot(1/\delta)^{1/(d-1)}))$ 和样本复杂度（即输入的数量）的 $-coin 问题) ${\mathop{\mathrm{poly}}}(1/\delta).$ 这与 O'Donnell 和 Wimmer [ICALP 2007: Automata, Languages and Programming, Lecture Notes in Comput. 科学。4596, Springer, New York, 2007, pp. 195--206] 和 Amano [ICALP 2009: Automata, Languages and Programming, Lecture Notes in Comput. 科学。5555, Springer, New York, 2009, pp. 59--70] 在尺寸方面（最佳），同时将样本复杂度从 $\exp(O(d\cdot(1/\delta)^{1/(d-1)}))$ 提高到 ${\mathop{\mathrm{poly}}}(1/ \delta)$。改进的样本复杂度对于证明大小层次定理至关重要。关于下限，我们表明即使对于明显更强的 ${\mathrm{AC}}^0[\oplus]$ 公式模型（也允许 NOT 和Parity gates）：形式上，我们证明了解决 $\delta$-coin 问题的任何 ${\mathrm{AC}}^0[\oplus]$ 公式必须具有大小 $\exp(\Omega(d\cdot(1 /\delta)^{1/(d-1)})).$ 这加强了 Shaltiel 和 Viola [SIAM J. Comput., 39 (2010), pp. 3122--3154] 的结果，他们证明了 $ \exp(\Omega((1/\delta)^{1/(d+2)}))$${\mathrm{AC}}^0[\oplus]$ 电路的下界，以及 Cohen 的结果、Ganor 和 Raz [近似随机，LIPIcs。莱布尼茨国际 过程 通知。28, Schloss Dagstuhl, Leibniz-Zentrum fuer Informatik, Wadern, 2014, pp. 618--629], 谁展示了 $\exp(\Omega((1/\delta)^{1/(d-1)}) )$ ${\mathrm{AC}}^0$ 电路的下限。上限是去随机化，涉及使用 Janson 不等式和基于经典多项式的组合设计的扩展。对于下限，我们证明了解决 $\delta$-coin 问题的 ${\mathbb{F}}_2$ 上的多元多项式的最优（最多为一个常数因子）度数下限，这可能是独立的兴趣。s 不等式和基于经典多项式的组合设计的扩展。对于下限，我们证明了解决 $\delta$-coin 问题的 ${\mathbb{F}}_2$ 上的多元多项式的最优（最多为一个常数因子）度数下限，这可能是独立的兴趣。s 不等式和基于经典多项式的组合设计的扩展。对于下限，我们证明了解决 $\delta$-coin 问题的 ${\mathbb{F}}_2$ 上的多元多项式的最优（最多为一个常数因子）度数下限，这可能是独立的兴趣。