用渠化深度表示布尔函数的分层与计数
Stratification and enumeration of Boolean functions by canalizing depth
摘要：
在过去的十几年里，布尔网络模型在计算系统生物学中越来越受欢迎。这些网络中的许多使用渠化布尔函数，这导致了对这些函数的研究的兴趣增加。一个函数的渠道化深度描述了有多少渠道化变量可以被递归地剔除，直到剩下一个非渠道化函数。在本文中，我们展示了如何每一个布尔函数有一个独特的代数形式，涉及扩展单项式层和一个定义良好的核心多项式。这概括了最近的工作嵌套canalizing函数的代数结构，它产生了分层的所有布尔函数的canalizing深度。其结果是，我们得到了封闭的公式的数量的n-变量布尔函数的深度为k，同时推广计数公式的渠化，嵌套渠化功能。
英文摘要：
Boolean network models have gained popularity in computational systems biology over the last dozen years. Many of these networks use canalizing Boolean functions, which has led to increased interest in the study of these functions. The canalizing depth of a function describes how many canalizing variables can be recursively picked off, until a non-canalizing function remains. In this paper, we show how every Boolean function has a unique algebraic form involving extended monomial layers and a well-defined core polynomial. This generalizes recent work on the algebraic structure of nested canalizing functions, and it yields a stratification of all Boolean functions by their canalizing depth. As a result, we obtain closed formulas for the number of n-variable Boolean functions with depth k, which simultaneously generalizes enumeration formulas for canalizing, and nested canalizing functions.
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