发表机构
University of Latvia（拉脱维亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究构造总布尔函数$G$，证明证书复杂度可达到1/3-近似度的四次方倍，改进了此前的三次方倍分离结果，通过cheat-sheet框架实现最优分离，核心是构造低度验证的近似多项式。
AI 中文摘要
我们证明，证书复杂度（记为$C$）可以比$1/3$-近似度（记为$\tilde{deg}$）大到四次方倍。更准确地说，我们构造了一族总布尔函数$G$，满足$C(G) = \tilde{\tilde{\tilde{deg}}}(G)^4$，其中$C$表示证书复杂度，$\tilde{deg}$表示$1/3$-近似度。该结果在多对数因子范围内是最优的，因为根据Nisan以及Nisan–Szegedy的经典块灵敏度界，每个总布尔函数$f$都满足$C(f)\tilde{deg}(f)^4$。因此，该结果缩小了这两个度量之间的差距，改进了Balodis、Ben-David、Göös、Jain和Kothari之前已知的最优分离结果$C(f)=\tilde{\tilde{\tilde{deg}}}(f)^3$。我们的构造从他们用于将0-证书复杂度与无歧义1-证书复杂度二次分离的部分函数出发，该部分函数已具备所需的证书硬度，但它的0-证书是非结构化的，阻碍了低度验证器的推导。我们保留其1-条件，并将0-输入限制为那些由结构化族认证的输入，该结构化族的有效性允许低度逼近，同时保持二次硬度。随后，将具有低度验证器的该部分函数通过cheat-sheet框架处理，得到具有所声称分离的总布尔函数$G$。主要技术要素是一个近似多项式，它以$\tilde{O}(\tilde{n})$的度数验证证书。该验证器形成与断言的0-证书兼容的候选1-证书的低度计数$W$，并测试该计数是否为零。关键在于，该构造确保$W$从未超过$\tilde{O}(n)$，而非其计数的$\tilde{\theta}(n^2)$候选对，从而将验证降至$\tilde{O}(\tilde{n})$的度数。
英文摘要
We prove that certificate complexity can be quartically larger than approximate degree. More precisely, we construct a family of total Boolean functions $G$ with $$ C(G) = \tildeΩ(\tilde{deg}(G)^4), $$ where $C$ denotes certificate complexity and $\tilde{deg}$ denotes $1/3$-approximate degree. This is optimal up to polylogarithmic factors, since every total Boolean function $f$ satisfies $C(f)\le O(\tilde{deg}(f)^4)$ by the classical block-sensitivity bounds of Nisan and Nisan--Szegedy. Thus the result closes the gap between these two measures and improves the previously best known separation $C(f)=\tildeΩ(\tilde{deg}(f)^3)$ by Balodis, Ben-David, Göös, Jain, and Kothari. The construction starts from the partial function they used to quadratically separate $0$-certificate complexity from unambiguous $1$-certificate complexity. It already has the required certificate hardness, but its $0$-certificates are unstructured, which blocks the derivation of a low-degree verifier. We keep its $1$-condition and restrict the $0$-inputs to those certified by a structured family whose validity admits a low-degree approximant, while preserving the quadratic hardness. The partial function with its low-degree verifier is then fed through the cheat-sheet framework to yield the total function $G$ with the claimed separation. The main technical ingredient is an approximate polynomial that verifies the certificate in degree $\tilde{O}(\sqrt n)$. The verifier forms a low-degree count $W$ of the candidate $1$-certificates that remain compatible with the asserted $0$-certificate, and tests whether this count is zero. Crucially, the construction ensures that $W$ never exceeds $\tilde{O}(n)$, instead of the $Θ(n^2)$ candidate pairs it counts bringing the verification down to degree $\tilde{O}(\sqrt n)$.
Comments17 pages
