Let $\mathsf{TH}_k$ denote the $k$-out-of-$n$ threshold function: given $n$ input Boolean variables, the output is $1$ if and only if at least $k$ of the inputs are $1$. We consider the problem of computing the $\mathsf{TH}_k$ function using noisy readings of the Boolean variables, where each reading is incorrect with some fixed and known probability $p \in (0,1/2)$. As our main result, we show that, when $k = o(n)$, it is both sufficient and necessary to use $$(1 \pm o(1)) \frac{n\log \frac{k}{\delta}}{D_{\mathsf{KL}}(p || 1-p)}$$ queries in expectation to compute the $\mathsf{TH}_k$ function with a vanishing error probability $\delta = o(1)$, where $D_{\mathsf{KL}}(p || 1-p)$ denotes the Kullback-Leibler divergence between $\mathsf{Bern}(p)$ and $\mathsf{Bern}(1-p)$ distributions. In particular, this says that $(1 \pm o(1)) \frac{n\log \frac{1}{\delta}}{D_{\mathsf{KL}}(p || 1-p)}$ queries in expectation are both sufficient and necessary to compute the $\mathsf{OR}$ and $\mathsf{AND}$ functions of $n$ Boolean variables. Compared to previous work, our result tightens the dependence on $p$ in both the upper and lower bounds.

翻译：设$\mathsf{TH}_k$表示$k$-out-of-$n$阈值函数：给定$n$个输入布尔变量，当且仅当至少$k$个输入为$1$时输出为$1$。我们考虑利用布尔变量的噪声读数计算$\mathsf{TH}_k$函数的问题，其中每次读数以固定且已知的概率$p \in (0,1/2)$出错。作为主要结果，我们证明：当$k = o(n)$时，为以渐近消失的错误概率$\delta = o(1)$计算$\mathsf{TH}_k$函数，期望查询次数达到$$(1 \pm o(1)) \frac{n\log \frac{k}{\delta}}{D_{\mathsf{KL}}(p || 1-p)}$$既是充分的也是必要的，其中$D_{\mathsf{KL}}(p || 1-p)$表示$\mathsf{Bern}(p)$与$\mathsf{Bern}(1-p)$分布之间的Kullback-Leibler散度。特别地，该结果表明：以$(1 \pm o(1)) \frac{n\log \frac{1}{\delta}}{D_{\mathsf{KL}}(p || 1-p)}$期望查询次数计算$n$个布尔变量的$\mathsf{OR}$和$\mathsf{AND}$函数既是充分的也是必要的。与已有工作相比，我们的结果在上下界中均强化了对参数$p$的依赖性。