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.
英文摘要：
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.
Degree and Sensitivity: tails of two distributions
布尔函数f的灵敏度是所有输入x上x的灵敏坐标数的最大值。著名的尼桑敏感性猜想（也见尼桑和塞格迪）指出，每个敏感性为s的布尔函数都可以通过次数为多的实数上的多项式来计算。然而，最著名的次数上界是指数的，而不是s中的多项式。我们的主要结果是猜想的一个近似版本：每个具有灵敏度s的布尔函数都可以用一个次数为O（slog（1/n））的多项式ε-逼近（在L_2中）。这是对《诗经》民俗学范畴的第一次改进。进一步，我们证明了对于任意d 0，将其上界提高到O（s^c log（1/n）^d）$< 1 and any c >将意味着敏感性猜想.因此，我们的结果基本上是人们在不证明猜想的情况下所能期望的最好结果。我们假设一个强大的模拟的敏感性猜想：如果大多数输入的布尔函数f具有低灵敏度，那么大多数的傅立叶质量的f是集中在小的子集，并提出了一种方法来证明这个猜想。
Decision Tree Complexity versus Block Sensitivity and Degree
Relations between the decision tree complexity and various other complexity measures of Boolean functions is a thriving topic of research in computational complexity . It is known that decision tree complexity is bounded above by the cube of block sensitivity, and the cube of polynomial degree. However, the widest separation between decision tree complexity and each of block sensitivity and degree that is witnessed by known Boolean functions is quadratic. In this work, we investigate the tightness of the existing cubic upper bounds. We improve the cubic upper bounds for many interesting classe…
Boolean functions with symmetry properties are interesting from a complexity theory perspective; extensive research has shown that these functions, if nonconstant, must have high `complexity' according to various measures. In recent work of this type, Sun gave bounds on the block sensitivity of nonconstant Boolean functions invariant under a transitive permutation group. Sun showed that all such functions satisfy bs(f) = Omega(N^{1/3}), and that there exists such a function for which bs(f) = O(N^{3/7}ln N). His example function belongs to a subclass of transitively invariant functions called t…
The estimates for Boolean surface area of polynomial threshold functions
Nankai University（南开大学）；Institute for Basic Science (IBS)（基础科学研究院）；University of Washington（华盛顿大学）；MSU（密歇根州立大学）；Max Planck Institute for Mathematics（马普数学研究所）；University of South Carolina（南卡罗来纳大学）
