We prove a $k^{-\Omega(\log(\varepsilon_2 - \varepsilon_1))}$ lower bound for adaptively testing whether a Boolean function is $\varepsilon_1$-close to or $\varepsilon_2$-far from $k$-juntas. Our results provide the first superpolynomial separation between tolerant and non-tolerant testing for a natural property of boolean functions under the adaptive setting. Furthermore, our techniques generalize to show that adaptively testing whether a function is $\varepsilon_1$-close to a $k$-junta or $\varepsilon_2$-far from $(k + o(k))$-juntas cannot be done with $\textsf{poly} (k, (\varepsilon_2 - \varepsilon_1)^{-1})$ queries. This is in contrast to an algorithm by Iyer, Tal and Whitmeyer [CCC 2021] which uses $\textsf{poly} (k, (\varepsilon_2 - \varepsilon_1)^{-1})$ queries to test whether a function is $\varepsilon_1$-close to a $k$-junta or $\varepsilon_2$-far from $O(k/(\varepsilon_2-\varepsilon_1)^2)$-juntas.

翻译：我们证明了在适应性测试中，布尔函数与$k$-Junta在$\varepsilon_1$-接近或$\varepsilon_2$-远离的判定问题具有$k^{-\Omega(\log(\varepsilon_2 - \varepsilon_1))}$的下界。该结果首次在自适应设置下，针对布尔函数的自然性质，实现了容忍测试与非容忍测试之间的超多项式分离。此外，我们的方法推广后表明，在适应性测试中，判定函数与$k$-Junta在$\varepsilon_1$-接近或与$(k + o(k))$-Junta在$\varepsilon_2$-远离的问题，无法通过$\textsf{poly} (k, (\varepsilon_2 - \varepsilon_1)^{-1})$次查询完成。这与Iyer、Tal和Whitmeyer [CCC 2021]的算法形成对比，该算法使用$\textsf{poly} (k, (\varepsilon_2 - \varepsilon_1)^{-1})$次查询来测试函数与$k$-Junta在$\varepsilon_1$-接近或与$O(k/(\varepsilon_2-\varepsilon_1)^2)$-Junta在$\varepsilon_2$-远离的判定。