基于硬币问题的AC $^0[\oplus]$的固定深度尺寸层次定理
A Fixed-Depth Size-Hierarchy Theorem for AC$^0[\oplus]$ via the Coin Problem
摘要：
本文证明了均匀AC $^0[\oplus]$电路的第一个固定深度尺寸层次定理;特别地，对于固定$d$，深度$d$和尺寸$n^k$的均匀AC $^0[\oplus]$公式的类$\mathcal{C}_{d,k}$形成无限层次。为此，我们找到了第一类显式函数，它给出了AC $^0[\oplus]$公式的匹配上界和下界（直到多项式因子），该公式来自$δ$ -硬币问题，该问题是区分硬币的概率为$(1+δ)/2$或$(1-δ)/2,$，其中$δ$是$0$的参数。我们研究了这个问题的复杂性，并在上界和下界上都取得了进展。上限。我们发现明确的单调AC $^0$公式解决$δ$ -硬币问题，具有深度$d$，大小$\exp(O(d(1/δ)^{1/(d-1)}))$，和样本复杂度多$(1/δ)$，对于常数$d\ge2$。这与O 'Donnell和Wimmer（ICALP 2007）和Amano（ICALP 2009）在大小方面的上界相匹配，并提高了样本复杂性。下限。即使对于AC $^0[\oplus]$公式的较强模型，其上界也几乎是紧的（允许NOT和Parity Gate）：任何解决$δ$ -硬币问题的AC $^0[\oplus]$公式必须具有大小$\exp(Ω(d(1/δ)^{1/(d-1)})).$这加强了Cohen、Ganor和Raz的结果（近似随机2014年），谁证明了AC $^0$的类似结果，以及Shaltiel和Viola的结果（SICOMP 2010），他们给出了一个超多项式较弱（仍然是指数）的下界。上界是一个去随机化，涉及使用詹森不等式（据我们所知，第一次这样使用的不等式）和经典的组合设计。对于下界，我们证明了一个最佳的（常数因子）度下界的多元多项式在$\mathbb{F}_2$解决$δ$ -硬币问题，这可能是独立的利益。
英文摘要：
We prove the first Fixed-depth Size-hierarchy Theorem for uniform AC$^0[\oplus]$ circuits; in particular, for fixed $d$, the class $\mathcal{C}_{d,k}$ of uniform AC$^0[\oplus]$ formulas of depth $d$ and size $n^k$ form an infinite hierarchy. For this, we find the first class of explicit functions giving (up to polynomial factor) matching upper and lower bounds for AC$^0[\oplus]$ formulas, derived from the $δ$-Coin Problem, the computational problem of distinguishing between coins that are heads with probability $(1+δ)/2$ or $(1-δ)/2,$ where $δ$ is a parameter going to $0$. We study this problem's complexity and make progress on both upper bounds and lower bounds. Upper bounds. We find explicit monotone AC$^0$ formulas solving the $δ$-coin problem, having depth $d$, size $\exp(O(d(1/δ)^{1/(d-1)}))$, and sample complexity poly$(1/δ)$, for constant $d\ge2$. This matches previous upper bounds of O'Donnell and Wimmer (ICALP 2007) and Amano (ICALP 2009) in terms of size and improves the sample complexity. Lower bounds. The upper bounds are nearly tight even for the stronger model of AC$^0[\oplus]$ formulas (which allow NOT and Parity gates): any AC$^0[\oplus]$ formula solving the $δ$-coin problem must have size $\exp(Ω(d(1/δ)^{1/(d-1)})).$ This strengthens a result of Cohen, Ganor and Raz (APPROX-RANDOM 2014), who prove a similar result for AC$^0$, and a result of Shaltiel and Viola (SICOMP 2010), who give a superpolynomially weaker (still exponential) lower bound. The upper bound is a derandomization involving a use of Janson's inequality (as far as we know, the first such use of the inequality) and classical combinatorial designs. For the lower bound, we prove an optimal (up to constant factor) degree lower bound for multivariate polynomials over $\mathbb{F}_2$ solving the $δ$-coin problem, which may be of independent interest.
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