We give a poly$(s,1/ε)$-query algorithm for testing whether an unknown and arbitrary function $f: \{0,1\}^n \to \{0,1\}$ is an $s$-term DNF, in the challenging relative-error framework for Boolean function property testing that was recently introduced and studied in a number of works [CDH+25b, CPPS25a, CPPS25b, CDH+25a]. This gives the first example of a rich and natural class of functions which may depend on a super-constant number of variables and yet is efficiently testable in the relative-error model with constant query complexity. A crucial new ingredient enabling our approach is a novel decomposition of any $s$-term DNF formula into ``local clusters'' of terms. Our results demonstrate that this new decomposition can be usefully exploited for algorithms even when the $s$-term DNF is not explicitly given; we believe that this decomposition may have applications in other contexts.

翻译：我们提出了一种查询复杂度为$\text{poly}(s,1/\varepsilon)$的算法，用于测试任意未知函数$f: \{0,1\}^n \to \{0,1\}$是否为$s$项析取范式（DNF）。该算法基于近期一系列工作[CDH+25b, CPPS25a, CPPS25b, CDH+25a]中引入并研究的布尔函数性质测试的相对误差框架，该框架具有显著挑战性。这一成果首次展示了一个丰富且自然的函数类——其可能依赖于超常数个变量——在相对误差模型中仍能以常数查询复杂度实现高效测试。我们方法的关键在于提出了一种新颖的分解技术，可将任意$s$项DNF公式分解为若干“局部簇”的合取项。我们的结果表明，即使$s$项DNF公式未显式给出，这种新分解仍能被算法有效利用；我们相信该分解技术在其他研究场景中也可能具有应用价值。