Monotonicity testing of Boolean functions on the hypergrid, $f:[n]^d \to \{0,1\}$, is a classic topic in property testing. Determining the non-adaptive complexity of this problem is an important open question. For arbitrary $n$, [Black-Chakrabarty-Seshadhri, SODA 2020] describe a tester with query complexity $\widetilde{O}(\varepsilon^{-4/3}d^{5/6})$. This complexity is independent of $n$, but has a suboptimal dependence on $d$. Recently, [Braverman-Khot-Kindler-Minzer, ITCS 2023] and [Black-Chakrabarty-Seshadhri, STOC 2023] describe $\widetilde{O}(\varepsilon^{-2} n^3\sqrt{d})$ and $\widetilde{O}(\varepsilon^{-2} n\sqrt{d})$-query testers, respectively. These testers have an almost optimal dependence on $d$, but a suboptimal polynomial dependence on $n$. In this paper, we describe a non-adaptive, one-sided monotonicity tester with query complexity $O(\varepsilon^{-2} d^{1/2 + o(1)})$, independent of $n$. Up to the $d^{o(1)}$-factors, our result resolves the non-adaptive complexity of monotonicity testing for Boolean functions on hypergrids. The independence of $n$ yields a non-adaptive, one-sided $O(\varepsilon^{-2} d^{1/2 + o(1)})$-query monotonicity tester for Boolean functions $f:\mathbb{R}^d \to \{0,1\}$ associated with an arbitrary product measure.

翻译：超网格上布尔函数 $f:[n]^d \to \{0,1\}$ 的单调性测试是性质检验中的一个经典课题。确定该问题的非适应复杂性是一个重要的开放性问题。对于任意 $n$，[Black-Chakrabarty-Seshadhri, SODA 2020] 描述了查询复杂度为 $\widetilde{O}(\varepsilon^{-4/3}d^{5/6})$ 的测试器。该复杂度与 $n$ 无关，但在 $d$ 上的依赖次优。最近，[Braverman-Khot-Kindler-Minzer, ITCS 2023] 和 [Black-Chakrabarty-Seshadhri, STOC 2023] 分别描述了 $\widetilde{O}(\varepsilon^{-2} n^3\sqrt{d})$ 和 $\widetilde{O}(\varepsilon^{-2} n\sqrt{d})$ 查询次数的测试器。这些测试器在 $d$ 上的依赖几乎最优，但在 $n$ 上的多项式依赖则是次优的。本文描述了一个非适应、单侧的单调性测试器，其查询复杂度为 $O(\varepsilon^{-2} d^{1/2 + o(1)})$，与 $n$ 无关。忽略 $d^{o(1)}$ 因子，我们的结果解决了超网格上布尔函数单调性测试的非适应复杂性。与 $n$ 的无关性为关联任意乘积测度的布尔函数 $f:\mathbb{R}^d \to \{0,1\}$ 提供了非适应、单侧的 $O(\varepsilon^{-2} d^{1/2 + o(1)})$ 查询次数的单调性测试器。