Boolean functions can be represented in many ways including logical forms, truth tables, and polynomials. Additionally, Boolean functions have different canonical representations such as minimal disjunctive normal forms. Other canonical representation is based on the polynomial representation of Boolean functions where they can be written as a nested product of canalizing layers and a polynomial that contains the noncanalizing variables. In this paper we study the problem of identifying the canalizing layers format of Boolean functions. First, we show that the problem of finding the canalizing layers is NP-hard. Second, we present several algorithms for finding the canalizing layers of a Boolean function, discuss their complexities, and compare their performances. Third, we show applications where the computation of canalizing layers can be used for finding a disjunctive normal form of a nested canalizing function. Another application deals with the reverse engineering of Boolean networks with a prescribed layering format. Finally, implementations of our algorithms in Python and in the computer algebra system Macaulay2 are available at https://github.com/ckadelka/BooleanCanalization.

中文翻译：

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揭示布尔函数的运河结构：算法和应用

布尔函数可以用多种方式表示，包括逻辑形式、真值表和多项式。此外，布尔函数具有不同的规范表示，例如最小析取范式。其他规范表示基于布尔函数的多项式表示，其中它们可以写为运河化层和包含非运河化变量的多项式的嵌套乘积。在本文中，我们研究了识别布尔函数的运河层格式的问题。首先，我们证明了寻找运河层的问题是 NP 难的。其次，我们提出了几种用于查找布尔函数的运河层的算法，讨论它们的复杂性，并比较它们的性能。第三，我们展示了一些应用，其中运河层的计算可用于寻找嵌套运河函数的析取范式。另一个应用程序处理具有规定分层格式的布尔网络的逆向工程。最后，我们的算法在 Python 和计算机代数系统 Macaulay2 中的实现可从 https://github.com/ckadelka/BooleanCanalization 获得。