Boolean function monotonicity testing requires (almost) $n^{1/2}$ queries

Abstract

We show that for any constant $c>0$, any (two-sided error) adaptive algorithm for testing monotonicity of Boolean functions must have query complexity $\hat{I}\text{c◯}(n^{1/2-c})$. This improves the $\tilde{\hat{I}}\text{c◯}(n^{1/3})$ lower bound of [CWX17] and almost matches the $\tilde{O}(\sqrt{n})$ upper bound of [KMS18].