Domain Reduction for Monotonicity Testing: A o(d) Tester for Boolean Functions in d-Dimensions

We describe a Õ(d5/6)-query monotonicity tester for Boolean functions f: [n]d → 0, 1 on the nhypergrid. This is the first o(d) monotonicity tester with query complexity independent of n. Motivated by this independence of n, we initiate the study of monotonicity testing of measurable Boolean functions f: ℝd → 0, 1 over the continuous domain, where the distance is measured with respect to a product distribution over ℝd. We give a Õ(d5/6)-query monotonicity tester for such functions. Our main technical result is a domain reduction theorem for monotonicity. For any function f: [n]d → 0, 1, let εf be its distance to monotonicity. Consider the restriction of the function on a random [k]d sub-hypergrid of the original domain. We show that for k = poly(d/εf), the expected distance of the restriction is . Previously, such a result was only known for d = 1 (Berman-Raskhodnikova-Yaroslavtsev, STOC 2014). Our result for testing Boolean functions over [n]d then follows by applying the d5/6 · poly(1/ε log n, log d)-query hypergrid tester of Black-Chakrabarty-Seshadhri (SODA 2018). To obtain the result for testing Boolean functions over ℝd, we use standard measure theoretic tools to reduce monotonicity testing of a measurable function f to monotonicity testing of a discretized version of f over a hypergrid domain [N]d for large, but finite, N (that may depend on f). The independence of N in the hypergrid tester is crucial to getting the final tester over ℝd.