K-Monotonicity is Not Testable on the Hypercube

We continue the study of k-monotone Boolean functions in the property testing model, initiated by Canonne et al. (ITCS 2017). A function f : {0, 1} → {0, 1} is said to be kmonotone if it alternates between 0 and 1 at most k times on every ascending chain. Such functions represent a natural generalization of (1-)monotone functions, and have been recently studied in circuit complexity, PAC learning, and cryptography. In property testing, the fact that 1-monotonicity can be locally tested with polyn queries led to a previous conjecture that k-monotonicity can be tested with poly(n) queries. In this work we disprove the conjecture, and show that even 2-monotonicity requires an exponential in √ n number of queries. Furthermore, even the apparently easier task of distinguishing 2-monotone functions from functions that are far from being n.01-monotone also requires an exponential number of queries. Our results follow from constructions of families that are hard for a canonical tester that picks a random chain and queries all points on it. Our techniques rely on a simple property of the violation graph and on probabilistic arguments necessary to understand chain tests. ∗Purdue University. Email: [email protected]. Research supported in part by NSF CCF-1649515. †Purdue University. Email: [email protected]. Research supported in part by NSF CCF-1649515 and NSF CCF-1618918. ‡Duquesne University. Email: [email protected].