We consider the problem of computing a function of $n$ variables using noisy queries, where each query is incorrect with some fixed and known probability $p \in (0,1/2)$. Specifically, we consider the computation of the $\mathsf{OR}$ function of $n$ bits (where queries correspond to noisy readings of the bits) and the $\mathsf{MAX}$ function of $n$ real numbers (where queries correspond to noisy pairwise comparisons). We show that an expected number of queries of \[ (1 \pm o(1)) \frac{n\log \frac{1}{\delta}}{D_{\mathsf{KL}}(p \| 1-p)} \] is both sufficient and necessary to compute both functions 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. Compared to previous work, our results tighten the dependence on $p$ in both the upper and lower bounds for the two functions.

翻译：我们研究了在噪声查询下计算 $n$ 个变量函数的问题，其中每次查询以固定且已知的概率 $p \in (0,1/2)$ 发生错误。具体而言，我们考虑了 $n$ 比特的 $\mathsf{OR}$ 函数（查询对应于比特的噪声读取）和 $n$ 个实数的 $\mathsf{MAX}$ 函数（查询对应于噪声成对比较）的计算。我们证明，对于这两个函数，当错误概率 $\delta = o(1)$ 时，期望查询次数为 \[ (1 \pm o(1)) \frac{n\log \frac{1}{\delta}}{D_{\mathsf{KL}}(p \| 1-p)} \] 既是充分也是必要的，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。与以往工作相比，我们的结果在上下界对参数 $p$ 的依赖关系上均进行了收紧。