AI 中文总结
本研究利用F₂多项式与多数函数异或的指数相关界，证明Almost-⊕P等于BP·⊕P，构建欺骗高次F₂多项式的伪随机生成器，并给出Toda定理前半部分的随机预言机证明。
AI 中文摘要
利用Chattopadhyay、Hatami、Lee、Lovett、Tal和Viola近期提出的$\text{F}_2$多项式与多数函数异或之间的指数相关界，我们证明了$\text{Almost-}\bigoplus\text{P} = \text{BP·}\bigoplus\text{P}$，其中$\text{Almost-}\bigoplus\text{P}$是指对于随机预言机$R$，以概率1属于$\bigoplus\text{P}^R$的语言类。这是Bennett和Gill提出的$\text{Almost-P}=\text{BPP}$、以及Nisan和Wigderson提出的$\text{Almost-PH}=\text{PH}$的奇偶性对应版本。证明的关键要素是一个种子长度为多项式的伪随机生成器，它能欺骗指数多变量上的多项式次数$\text{F}_2$多项式。作为应用，我们遵循Regan和Royer的方法，完成了Toda定理前半部分$\text{PH}\subseteq \text{BP·}\bigoplus\text{P}$的随机预言机证明。在随机预言机下，通过逐层应用Valiant-Vazirani定理和Papadimitriou-Zachos定理，多项式时间谱系会坍缩到$\bigoplus\text{P}$中，且全程无需将概率量词穿过预言机。我们的结果随后移除了预言机的限制。我们还将该论证与Fortnow（2009）给出的Toda定理简单证明进行了对比。
英文摘要
Using the recent exponential correlation bounds of Chattopadhyay, Hatami, Lee, Lovett, Tal and Viola between $\mathbb{F}_2$-polynomials and the XOR of majorities, we show that $\mathrm{Almost}\text{-}\oplus\mathrm{P} = \mathrm{BP}\cdot\oplus\mathrm{P}$, where $\mathrm{Almost}\text{-}\oplus\mathrm{P}$ is the class of languages that lie in $\oplus\mathrm{P}^R$ with probability one for a random oracle $R$. This is the parity analogue of Bennett and Gill's $\mathrm{Almost}\text{-}\mathrm{P} = \mathrm{BPP}$ and Nisan and Wigderson's $\mathrm{Almost}\text{-}\mathrm{PH} = \mathrm{PH}$. The key ingredient is a pseudorandom generator with polynomial seed length that fools $\mathbb{F}_2$-polynomials of polynomial degree on exponentially many variables. As an application we complete a random-oracle proof of the first half of Toda's theorem, $\mathrm{PH} \subseteq \mathrm{BP}\cdot\oplus\mathrm{P}$, following an approach of Regan and Royer. Relative to a random oracle, the polynomial hierarchy collapses into $\oplus\mathrm{P}$ by applying Valiant-Vazirani and Papadimitriou-Zachos level by level, with no probabilistic quantifier ever moved through an oracle. Our result then removes the oracle. We compare this argument with the simple proof of Toda's theorem by Fortnow (2009).
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