Testing $k$-Monotonicity: The Rise and Fall of Boolean Functions

A Boolean k-monotone function defined over a finite poset domain D alternates between the values 0 and 1 at most k times on any ascending chain in D. Therefore, k-monotone functions are natural generalizations of the classical monotone functions, which are the 1-monotone functions. Motivated by the recent interest in k-monotone functions in the context of circuit complexity and learning theory, and by the central role that monotonicity testing plays in the context of property testing, we initiate a systematic study of k-monotone functions, in the property testing model. In this model, the goal is to distinguish functions that are k-monotone (or are close to being k-monotone) from functions that are far from being k-monotone. Our results include the following. 1. We demonstrate a separation between testing k-monotonicity and testing monotonicity, on the hypercube domain {0,1}(d), for k >= 3; 2. We demonstrate a separation between testing and learning on {0,1}(d), for k = omega(log d): testing k-monotonicity can be performed with exp(O(root d . log d . log(1/epsilon))) queries, while learning k-monotone functions requires exp(Omega(k . root d . 1/epsilon)) queries (Blais et al. (RANDOM 2015)); 3. We present a tolerant test for k-monotonicity of functions f : [n](d) -> {0,1} with complexity independent of n. The test implies a tolerant test for monotonicity of functions f : [n](d) -> [0,1] in l(1) distance, which makes progress on a problem left open by Berman et al. (STOC 2014). Our techniques exploit the testing-by-learning paradigm, use novel applications of Fourier analysis on the grid [n](d), and draw connections to distribution testing techniques.