一种改进的完全性听写测验
An Improved Dictatorship Test with Perfect Completeness
摘要：
一个布尔函数$f:\{0,1\}^n\rightarrow \{0,1\}$被称为独裁者，如果它只依赖于一个变量，即$i\in [n]$的$f(x_1, x_2, \ldots, x_n) = x_i$。在这项工作中，我们研究了一个$k$查询独裁测试。独裁测试是证明约束满足问题的许多困难结果的核心。如果独裁测试接受任何独裁者函数，则称其具有{ \em完美完整性}。测试的{ \em可靠性}是它接受远离独裁者的任何函数的最大概率。我们的主要结果是一个$k$查询独裁测试与完美的完整性和健全性$ \frac{2k + 1}{2^k}$，其中$k$是形式$2^t -1$的任何整数$t > 2$。这改进了\cite{TY15}的结果，\cite{TY15}给出了具有稳健性$ \frac{2k + 3}{2^k}$的独裁测试。
英文摘要：
A Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ is called a dictator if it depends on exactly one variable i.e $f(x_1, x_2, \ldots, x_n) = x_i$ for some $i\in [n]$. In this work, we study a $k$-query dictatorship test. Dictatorship tests are central in proving many hardness results for constraint satisfaction problems. The dictatorship test is said to have {\em perfect completeness} if it accepts any dictator function. The {\em soundness} of a test is the maximum probability with which it accepts any function far from a dictator. Our main result is a $k$-query dictatorship test with perfect completeness and soundness $ \frac{2k + 1}{2^k}$, where $k$ is of the form $2^t -1$ for any integer $t > 2$. This improves upon the result of \cite{TY15} which gave a dictatorship test with soundness $ \frac{2k + 3}{2^k}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
