任意次数的初等对称布尔函数的证书复杂度

中文摘要 AI 辅助

设$\sigma_{n,d}$表示$n$个变量、次数为$d$的初等对称布尔函数。我们之前的工作确定了当$d$为奇数以及当$d$为2的幂时其证书复杂度，而二进制表示中至少有两个非零位的偶数次数仍未解决。我们解决了这个开放问题，并确定了所有$1\le d\le n$的$C(\sigma_{n,d})$。

英文摘要

Let $σ_{n,d}$ denote the elementary symmetric Boolean function of $n$ variables and degree $d$. Previous work determined $C(σ_{n,d})$ when $d$ is odd or a power of two, but the general even non-power-of-two case remained open. We determine the certificate complexity for every degree $1\le d\le n$, thereby completing the classification for elementary symmetric Boolean functions. Writing $d=2^t m$ with $m$ odd, we obtain an explicit formula in which the possible deficit from the maximal value $n$ is controlled by $2^t$, while the exact value is determined by a binary containment condition involving $m$. In particular, \[ n-2^{ν_2(d)}+1\le C(σ_{n,d})\le n, \] and we characterize when the upper bound is attained. Moreover, we determine the least positive period of the certificate-complexity deficit $Δ_d(n)=n-C(σ_{n,d})$: it is $1$ for odd $d$, equals $d$ when $d$ is a power of two, and equals $2^{\lfloor\log_2 d\rfloor+1}$ for even non-power-of-two $d$. This least-period problem is distinct from the classical periodicity of the underlying value sequence $\binom{j}{d}\bmod 2$. The known odd-degree and power-of-two formulas are recovered as special cases.

发表机构

• Governors State University
• Winston–Salem State University（温斯顿-塞勒姆州立大学）

机构由 AI 辅助整理，请以论文原文为准。