This paper focuses on the study of certain classes of Boolean functions that have appeared in several different contexts. Nested canalyzing functions have been studied recently in the context of Boolean network models of gene regulatory networks. In the same context, polynomial functions over finite fields have been used to develop network inference methods for gene regulatory networks. Finally, unate cascade functions have been studied in the design of logic circuits and binary decision diagrams. This paper shows that the class of nested canalyzing functions is equal to that of unate cascade functions. Furthermore, it provides a description of nested canalyzing functions as a certain type of Boolean polynomial function. Using the polynomial framework one can show that the class of nested canalyzing functions, or, equivalently, the class of unate cascade functions, forms an algebraic variety which makes their analysis amenable to the use of techniques from algebraic geometry and computational algebra. As a corollary of the functional equivalence derived here, a formula in the literature for the number of unate cascade functions provides such a formula for the number of nested canalyzing functions.

中文翻译：

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嵌套分析、Un​​ate 级联和多项式函数。

本文重点研究出现在几个不同上下文中的某些类别的布尔函数。最近在基因调控网络的布尔网络模型的背景下研究了嵌套分析功能。在相同的背景下，有限域上的多项式函数已被用于开发基因调控网络的网络推理方法。最后，在逻辑电路和二元决策图的设计中研究了 unate 级联函数。本文表明嵌套分析函数的类等于 unate 级联函数的类。此外，它还将嵌套分析函数描述为某种类型的布尔多项式函数。使用多项式框架可以证明嵌套分析函数的类，或者等价地，unate 级联函数类形成了一个代数变体，这使得它们的分析适合使用来自代数几何和计算代数的技术。作为此处导出的功能等价的推论，文献中的 unate 级联函数数量的公式提供了嵌套分析函数数量的公式。