嵌套催变函数的对称性质
Symmetry Properties of Nested Canalyzing Functions
AI总结：
本研究揭示了嵌套催变函数与对称布尔函数的关系，提出了对称性判定方法及算法，并推导了强不对称NCF数量的闭式表达式。
摘要：
许多研究者已经研究了各种布尔函数的对称性质。一类被称为嵌套催变函数（NCFs）的布尔函数已被用于模拟某些生物现象。我们发现了NCFs、对称布尔函数以及对称布尔函数的一种推广（我们称之为$r$-对称函数，其中$r$为对称级别）之间的一些有趣关系。利用NCFs的归一化表示，我们刻画了NCF的两个变量何时对称。利用这一刻画，我们表明给定$f$的标准表示，NCF $f$的对称级别可以轻松计算。我们还提出了一种算法，用于测试给定的$r$-对称函数是否为NCF。此外，我们表明对于任何具有$n$个变量的NCF $f$，文献中考虑的强不对称概念等价于$f$是$n$-对称的性质。我们利用这一结果推导出具有$n$个变量且为NCF且强不对称的布尔函数数量的闭式表达式。我们还识别了所有既是NCF又是对称的布尔函数。
英文摘要：
Many researchers have studied symmetry properties of various Boolean functions. A class of Boolean functions, called nested canalyzing functions (NCFs), has been used to model certain biological phenomena. We identify some interesting relationships between NCFs, symmetric Boolean functions and a generalization of symmetric Boolean functions, which we call $r$-symmetric functions (where $r$ is the symmetry level). Using a normalized representation for NCFs, we develop a characterization of when two variables of an NCF are symmetric. Using this characterization, we show that the symmetry level of an NCF $f$ can be easily computed given a standard representation of $f$. We also present an algorithm for testing whether a given $r$-symmetric function is an NCF. Further, we show that for any NCF $f$ with $n$ variables, the notion of strong asymmetry considered in the literature is equivalent to the property that $f$ is $n$-symmetric. We use this result to derive a closed form expression for the number of $n$-variable Boolean functions that are NCFs and strongly asymmetric. We also identify all the Boolean functions that are NCFs and symmetric.
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