AI 中文总结
该研究解决块敏感性与谱敏感性的关系问题，构造出块敏感性超过谱敏感性平方的布尔函数，证明块敏感性可突破二次上界，相关结果已在Lean中形式化验证。
AI 中文摘要
布尔函数的谱敏感性λ(f)是其敏感性图邻接矩阵的最大特征值，它是所有标准查询复杂度度量的下界，引入该概念的Aaronson、Ben-David、Kothari、Rao和Tal提出问题：块敏感性是否至多为其平方，即bs(f)=O(λ(f)²)？我们证明这并不成立。我们构造了一个含2017584个变量的完全布尔函数，其bs(f)≥14011且λ(f)≤89.0162，使得bs(f)≥λ(f)^2.127，进而通过组合得到一族函数，满足λ(fₙ)→∞且bs(fₙ)=Ω(λ(fₙ)^2.127)。该函数是由双正则锦标赛顶点索引的k个子立方体的并集的指示函数，构造中的自由度由Lovász局部引理确定。主要结果已在Lean中形式化验证。我们还给出数值证据：同一家族中含1255个变量的函数达到接近2.20的指数，另有含30个变量的函数，其指数已超过2且谱敏感性可精确计算。
英文摘要
The spectral sensitivity $λ(f)$ of a Boolean function is the largest eigenvalue of the adjacency matrix of its sensitivity graph. It lower-bounds every standard measure of query complexity, and Aaronson, Ben-David, Kothari, Rao and Tal, who introduced it, asked whether block sensitivity is at most quadratic in it: is $bs(f)=O(λ(f)^{2})$? We show that it is not. We construct a total Boolean function on $2017584$ variables with $bs(f)\ge 14011$ and $λ(f)\le 89.0162$, so that $bs(f)\geλ(f)^{2.127}$, and hence by composition a family with $λ(f_n)\to\infty$ and $bs(f_n)=Ω(λ(f_n)^{2.127})$. The function is the indicator of a union of $k$ subcubes indexed by the vertices of a doubly regular tournament, and the freedom left in the construction is fixed by the Lovász local lemma. The main result has been formally verified in Lean. We also give numerical evidence that a member of the same family on $1255$ variables reaches an exponent near $2.20$, and exhibit a member on $30$ variables whose exponent already exceeds $2$ and whose spectral sensitivity can be computed exactly.
Comments15 pages
