A Boolean k -monotone function defined over a finite poset domain alternates between the values 0 and 1 at most k times on any ascending chain in . 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:

- We demonstrate a separation between testing k -monotonicity and testing monotonicity, on the hypercube domain 0 1 d , for k 3 ; - We demonstrate a separation between testing and learning on 0 1 d , for k = ( log d ) : testing k -monotonicity can be performed with 2 O ( d log d log 1 \eps ) queries, while learning k -monotone functions requires 2 ( k d 1 \eps ) queries (Blais et al. (RANDOM 2015)). - We present a tolerant test for functions f : [ n ] d 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 ] d , and draw connections to distribution testing techniques.