A number of complexity measures for Boolean functions have previously been introduced. These include (1) sensitivity, (2) block sensitivity, (3) witness complexity, (4) subcube partition complexity and (5) algorithmic complexity. Each of these is concerned with "worst-case" inputs. It has been shown that there is "asymptotic separation" between these complexity measures and very recently, due to the work of Huang, it has been established that they are all "polynomially related". In this paper, we study the notion of distributional complexity where the input bits are independent and one considers all of the above notions in expectation. We obtain a number of results concerning distributional complexity measures, among others addressing the above concepts of "asymptotic separation" and being "polynomially related" in this context. We introduce a new distributional complexity measure, local witness complexity, which only makes sense in the distributional context and we also study a new version of algorithmic complexity which involves partial information. Many interesting examples are presented including some related to percolation. The latter connects a number of the recent developments in percolation theory over the last two decades with the study of complexity measures in theoretical computer science.

翻译：此前已引入多种布尔函数的复杂度度量。这些度量包括：(1) 敏感度，(2) 块敏感度，(3) 见证复杂度，(4) 子立方体划分复杂度，以及 (5) 算法复杂度。这些度量均关注"最坏情况"输入。已有研究表明这些复杂度度量之间存在"渐近分离"，而最近由于黄皓的工作，已证实它们都是"多项式相关"的。本文研究分布复杂度的概念，其中输入位相互独立，并在期望意义下考察上述所有概念。我们获得了关于分布复杂度度量的若干结果，特别探讨了在此背景下的"渐近分离"和"多项式相关"概念。我们引入了一种新的分布复杂度度量——局部见证复杂度，该度量仅在分布背景下具有意义；同时我们还研究了一种涉及部分信息的新版本算法复杂度。本文提出了许多有趣的示例，包括一些与渗流相关的案例。后者将过去二十年间渗流理论的多项新进展与理论计算机科学中的复杂度度量研究联系起来。