X. Chen and R. Servedio and L.-Y. Tan.

In

**Lower bound:**
We prove an \Omega(n^{1/5}) lower bound on
the query complexity of any non-adaptive two-sided error
algorithm for testing whether an unknown Boolean function
f is monotone versus constant-far from monotone.
This gives an exponential improvement on the previous
lower bound of \Omega(log n) due to Fischer et al.. We show
that the same lower bound holds for monotonicity testing
of Boolean-valued functions over hypergrid domains
\{1,...,m\}^n for all m \geq 2.

**Upper bound:**
We present an \tilde{O}(n^{5/6})\poly(1/\epsilon)-query
algorithm that tests whether an unknown Boolean function
f is monotone versus \epsilon-far from monotone. Our algorithm,
which is non-adaptive and makes one-sided error, is a
modified version of the algorithm of Chakrabarty and
Seshadhri [2], which makes \tilde{O}(n^{7/8})\poly(1/\epsilon) queries.

pdf of conference version