AI 中文总结
本文完全确定了任意次基本对称布尔函数的敏感性、平均敏感性和块敏感性，并分类了其与证书复杂度的三种关系模式，给出了严格层级的充要条件。
AI 中文摘要
设 $\sigma_{n,d}$ 表示 $n$ 个变量、次数为 $d$ 的基本对称布尔函数。我们完全确定了 $\sigma_{n,d}$ 对每个 $1\le d\le n$ 的敏感性、平均敏感性和块敏感性。利用 Lucas 定理，我们获得了汉明重量值序列的均匀二进制描述，由此得出敏感性和平均敏感性公式，并将块敏感性的计算简化为至多四个显式候选值。将这些结果与证书复杂度的任意次公式相结合，我们确定了敏感性、块敏感性和证书复杂度之间的精确关系。我们还证明了对称布尔函数的一个一般结果：每个非常数对称布尔函数 $f$ 满足 \\[ \bs(f)\le \max\{s(f),C(f)-1\}. \\] 因此，对于非常数对称布尔函数，只能出现三种模式：\\[ s=\bs=C,\qquad s=\bs<C,\qquad s<\bs<C. \\] 对于基本对称布尔函数，我们给出了这三种模式各自成立的充要条件，从而完全分类了 $s(\sigma_{n,d})$、$\bs(\sigma_{n,d})$ 和 $C(\sigma_{n,d})$ 之间的关系。特别地，我们得到了完全严格层级 \\[ s(\sigma_{n,d})<\bs(\sigma_{n,d})<C(\sigma_{n,d}) \\] 的充要刻画，并展示了满足该层级的无限族。
英文摘要
Let $σ_{n,d}$ denote the elementary symmetric Boolean function of $n$ variables and degree $d$. We completely determine the sensitivity, average sensitivity, and block sensitivity of $σ_{n,d}$ for every $1\le d\le n$. Using Lucas' theorem, we obtain a uniform binary description of the Hamming-weight value sequence, from which the sensitivity and average-sensitivity formulas follow and the computation of block sensitivity reduces to at most four explicit candidates. Combining these results with the arbitrary-degree formula for certificate complexity, we determine the exact relations among sensitivity, block sensitivity, and certificate complexity. We also prove a general result for symmetric Boolean functions: every nonconstant symmetric Boolean function $f$ satisfies \[ \bs(f)\le \max\{s(f),C(f)-1\}. \] Consequently, only the three patterns \[ s=\bs=C,\qquad s=\bs<C,\qquad s<\bs<C \] can occur for nonconstant symmetric Boolean functions. For elementary symmetric Boolean functions, we give necessary and sufficient conditions for each of these three patterns, thereby completely classifying the relations among $s(σ_{n,d})$, $\bs(σ_{n,d})$, and $C(σ_{n,d})$. In particular, we obtain a necessary and sufficient characterization of the full strict hierarchy \[ s(σ_{n,d})<\bs(σ_{n,d})<C(σ_{n,d}), \] and exhibit infinite families for which it holds.
Comments26 pages
