P. Gopalan and N. Nisan and R. Servedio and K. Talwar and A. Wigderson.

A natural measure of smoothness of a Boolean function is its sensitivity (the largest number of Hamming neighbors of a point which differ from it in function value). The structure of smooth or equivalently low-sensitivity functions is still a mystery. A well-known conjecture states that every such Boolean function can be computed by a shallow decision tree. While this conjecture implies that smooth functions are easy to compute in the simplest computational model, to date no non-trivial upper bounds were known for such functions in any computational model, including unrestricted Boolean circuits. Even a bound on the description length of such functions better than the trivial 2^n does not seem to have been known.

In this work, we establish the first computational upper bounds on smooth Boolean functions:

1) We show that every sensitivity s function is uniquely specified by its values on a Hamming ball of radius 2s. We use this to show that such functions can be computed by circuits of size n^{O(s)}.

2) We show that sensitivity s functions satisfy a strong pointwise noise-stability guarantee for random noise of rate O(1/s). We use this to show that these functions have formulas of depth O(s log n).

3) We show that sensitivity s functions can be (locally) self-corrected from worst-case noise of rate exp(−O(s)).

All our results are simple, and follow rather directly from (variants of) the basic fact that that the function value at few points in small neighborhoods of a given point determine its function value via a majority vote. Our results confirm various consequences of the conjecture. They may be viewed as providing a new form of evidence towards its validity, as well as new directions towards attacking it.