TR17-088 Authors: Elena Grigorescu, Akash Kumar, Karl Wimmer

Publication: 11th May 2017 18:52

Downloads: 217

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We continue the study of $k$-monotone Boolean functions in the property testing model, initiated by Canonne et al. (ITCS 2017). A function $f:\{0,1\}^n\rightarrow \{0,1\}$ is said to be $k$-monotone 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 $\poly n$ queries led to a previous conjecture that $k$-monotonicity can be tested with $poly(n^k)$ queries. In this work we disprove the conjecture, and show that even $2$-monotonicity requires an exponential in $\sqrt{n}$ 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.