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}, for k ≥ 3; 2. We demonstrate a separation between testing and learning on {0, 1}, for k = ω(log d): testing k-monotonicity can be performed with 2O( √ d·log d·log 1/ε) queries, while learning k-monotone functions requires 2Ω(k· √ d·1/ε) queries (Blais et al. (RANDOM 2015)). 3. We present a tolerant test for functions f : [n] → {0, 1} with complexity independent of n, 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], and draw connections to distribution testing techniques. ∗Columbia University. Email: [email protected]. Research supported by NSF CCF-1115703 and NSF CCF-1319788. †Purdue University. Email: [email protected]. Research supported in part by NSF CCF-1649515. ‡Courant Institute of Mathematical Sciences, New York University. Email: [email protected]. §Purdue University. Email: [email protected]. ¶Duquesne University. Email: [email protected].