Smooth Boolean Functions are Easy: Efficient Algorithms for Low-Sensitivity Functions

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 2n does not seem to have been known. In this work, we establish the first computational upper bounds on smooth Boolean functions: 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 nO(s)}. 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). 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 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.