发表机构
Institute for Advanced Study（高等研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种完全自顶向下的新方法，证明奇偶校验函数对任意常深度 De Morgan 电路需要 $2^{n^{\Omega(1)}}$ 大小，并通过关键引理将下界改进至本质上紧的 $\exp(\epsilon_d n^{1/(d-1)})$。
AI 中文摘要
我们证明奇偶校验函数需要大小为 $2^{n^{\Omega(1)}}$ 的常深度 De Morgan 电路，采用了一种在 [HJP95] 意义上完全“自顶向下”的新方法。该证明关键依赖于先前一系列工作 [HJP95, PPZ99, MW19, GRSS24] 中发展的核心思想，这些工作此前建立了深度 3 和 4 电路的自顶向下下界。我们首先给出下界 $\exp(n^{3^{-d}})$ 的证明。在这种情况下，证明所需的相关组合思想几乎都已以某种形式存在于 [GRSS24] 中。然后我们提出该论证的两个扩展：第一个扩展达到下界 $\exp(\epsilon_d n^{1/(2d-2)})$，其中 $\epsilon_d>0$ 仅依赖于 $d$；第二个扩展达到本质上紧的下界 $\exp(\epsilon_d n^{1/(d-1)})$。这些改进的结果均依赖于建立一个关键引理，该引理量化了 $\{0,1\}^n$ 中的高熵随机变量在投影到随机小坐标集 $R \subseteq [n]$ 后，与均匀分布接近的程度。
英文摘要
We prove that Parity requires $2^{n^{Ω(1)}}$ size De Morgan circuits of constant depth using a new method which is completely "top-down" in the sense of [HJP95]. The proof relies crucially on the core ideas developed in a line of work [HJP95, PPZ99, MW19, GRSS24] which previously established top-down lower bounds for circuits of depth 3 and 4. We first present a proof of a lower bound $\exp(n^{3^{-d}})$. In this case, nearly all of the relevant combinatorial ideas necessary for the proof are already present in some form in [GRSS24]. We then present two extensions of this argument, the first achieving a lower bound $\exp(ε_d n^{1/(2d-2)})$ for some $ε_d>0$ depending only on $d$, and the second achieving the essentially tight lower bound $\exp(ε_d n^{1/(d-1)})$. These improved results each hinge on establishing a key lemma which quantifies the extent to which a high entropy random variable in $\{0,1\}^n$ will look close to uniform after projecting it onto a random small set of coordinates $R \subseteq [n]$.
Comments20 pages
