A Lower Bound for Nonadaptive, One-Sided Error Testing of Unateness of Boolean Functions over the Hypercube

Abstract: A Boolean function $f:{0,1}^d \mapsto {0,1}$ is unate if, along each coordinate, the function is either nondecreasing or nonincreasing. In this note, we prove that any nonadaptive, one-sided error unateness tester must make $\Omega(\frac{d}{\log d})$ queries. This result improves upon the $\Omega(\frac{d}{\log^2 d})$ lower bound for the same class of testers due to Chen et al. (STOC, 2017).