布尔函数的分布复杂性度量研究
A study of distributional complexity measures for Boolean functions
AI总结：
本文研究了布尔函数在独立输入下的分布复杂性度量，探讨了各度量在期望意义下的“渐近分离”与“多项式相关”性，并引入了局部见证复杂性与涉及部分信息的新版算法复杂性，还通过渗流例子建立了与理论计算机科学的联系。
摘要：
此前已经引入了许多关于布尔函数的复杂性度量。这些包括 (1) 敏感度，(2) 块敏感度，(3) 见证复杂性，(4) 子立方体划分复杂性和 (5) 算法复杂性。这些度量都关注“最坏情况”输入。已经证明，这些复杂性度量之间存在“渐近分离”，并且最近由于 Huang 的工作，已经确定它们都是“多项式相关”的。在本文中，我们研究了分布复杂性的概念，其中输入比特是独立的，并且人们考虑上述所有概念的期望。我们获得了关于分布复杂性度量的若干结果，其中包括在此背景下探讨上述“渐近分离”和“多项式相关”的概念。我们引入了一种新的分布复杂性度量，局部见证复杂性，它仅在分布背景下才有意义，并且我们还研究了涉及部分信息的新版本算法复杂性。文中给出了许多有趣的例子，包括一些与渗流相关的例子。后者将过去二十年中渗流理论的若干最新发展与理论计算机科学中复杂性度量的研究联系起来。
英文摘要：
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.
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