发表机构
Johns Hopkins University; Institute for Advanced Study（约翰斯·霍普金斯大学; 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
本文证明每个半比特生成器包含指数多个证明复杂度生成器，利用Pajor引理，并引入零误差分散器族，从几乎非平凡的半比特生成器构造证明复杂度生成器，展示范围回避问题的困难性。
AI 中文摘要
对于命题证明系统 $\mathcal{P}$ 和多项式时间函数 $G: \{0, 1\}^n \to \{0, 1\}^N$（其中 $N > 10n$），如果 $\mathcal{P}$ 无法高效证明对于每个 $y \in \{0, 1\}^N$ 的（适当编码的）陈述“$y\not\in\mathrm{Range}(G)$”，则称 $G$ 是针对 $\mathcal{P}$ 的*证明复杂度生成器*；如果 $\mathcal{P}$ 无法高效证明对于 $y \in \{0, 1\}^N$ 中不可忽略比例的陈述“$y \not\in\mathrm{Range}(G)$”，则称 $G$ 是针对 $\mathcal{P}$ 的*半比特生成器*。从定义可以看出，证明复杂度生成器在性质上强于半比特生成器。我们的主要结果是，或许与直觉相反，每个半比特生成器“包含”指数多个证明复杂度生成器。事实上，半比特生成器输出比特的随机子集以常数概率构成一个证明复杂度生成器。这一结果是 Pajor 引理（著名的 Sauer--Shelah 引理的加强版）的一个极其简单的推论。该结果使我们能够在若干新的、受关注的受限设置中展示范围回避问题（$\text{Avoid}$）的困难性。在此过程中，我们引入了*零误差分散器族*的概念，它可以将半比特生成器转化为证明复杂度生成器，并证明该族可以通过投影（即不增加任何电路复杂度开销）来计算。使用该族的另一种实例化，我们从*几乎非平凡*的半比特生成器 $G: \{0, 1\}^n \to \{0, 1\}^N$ 构造证明复杂度生成器，其中形如“$y \not \in \mathrm{Range}(G)$”的难证明陈述的数量仅略微超过 $2^n$（即该形式*假*陈述的数量）。
英文摘要
For a propositional proof system $\mathcal{P}$ and a polynomial-time function $G: \{0, 1\}^n \to \{0, 1\}^N$ ($N > 10n$), we say that $G$ is a *proof complexity generator* against $\mathcal{P}$ if $\mathcal{P}$ cannot efficiently prove the (suitably encoded) statement "$y\not\in\mathrm{Range}(G)$" for every $y \in \{0, 1\}^N$, and $G$ is a *demi-bits generator* against $\mathcal{P}$ if $\mathcal{P}$ cannot efficiently prove the statement "$y \not\in\mathrm{Range}(G)$" for a noticeable fraction of $y \in \{0, 1\}^N$. As can be seen from the definitions, proof complexity generators are qualitatively stronger objects than demi-bits generators. Our main result is that, perhaps counter-intuitively, every demi-bits generator "contains" exponentially many proof complexity generators. In fact, a random subset of output bits of a demi-bits generator forms a proof complexity generator with constant probability. This result is an extremely simple corollary of Pajor's Lemma (a strengthening of the well-known Sauer--Shelah Lemma). This result allows us to exhibit the hardness of the Range Avoidance problem ($\text{Avoid}$) in several new, restricted settings of interest. Along the way, we introduce the notion of *zero-error disperser families* that can transform demi-bits generators into proof complexity generators, and show that this family can be computed by projections (i.e., without any circuit complexity overhead). Using a different instantiation of this family, we construct proof complexity generators from *barely non-trivial* demi-bits generators $G: \{0, 1\}^n \to \{0, 1\}^N$, where the number of hard-to-prove statements of the form "$y \not \in \mathrm{Range}(G)$" just slightly exceeds $2^n$ (which is the number of *false* statements of this form).
CommentsAbstract shortened due to constraints
