嵌套C分析函数及其平均灵敏度
Nested Canalyzing Functions And Their Average Sensitivities
摘要：
本文通过得到嵌套解析函数唯一的代数范式（多项式型），给出了嵌套解析函数的完全刻画。我们引入了一个新的概念，层号NCF。在此基础上，我们得到了下列重要参数的显式表达式：1）所有嵌套分析函数的个数，2）所有给定层数的NCF的个数，3）任意NCF的汉明重量，4）任意NCF的任意变量的活动数，5）任意NCF的平均灵敏度。基于这些公式，我们证明了这些变量在外层的活动数较大，而在同一层中的活动数相等。当NCF只有一层时，平均灵敏度达到最小值。我们还证明了任何NCF的平均灵敏度（无论它有多少个变量）在0和2之间。因此，从理论上讲，我们证明了为什么NCF是稳定的，因为一个随机布尔函数具有平均灵敏度$\frac{n}{2}$。最后，我们推测，如果NCF具有最大的层数$n-1$，则NCF达到最大的平均灵敏度。因此，我们猜测任何NCF的平均灵敏度的统一上限可以减少到4/3，这是紧的。
英文摘要：
In this paper, we obtain complete characterization for nested canalyzing functions (NCFs) by obtaining its unique algebraic normal form (polynomial form). We introduce a new concept, LAYER NUMBER for NCF. Based on this, we obtain explicit formulas for the the following important parameters: 1) Number of all the nested canalyzing functions, 2) Number of all the NCFs with given LAYER NUMBER, 3) Hamming weight of any NCF, 4) The activity number of any variable of any NCF, 5) The average sensitivity of any NCF. Based on these formulas, we show the activity number is greater for those variables in out layer and equal in the same layer. We show the average sensitivity attains minimal value when the NCF has only one layer. We also prove the average sensitivity for any NCF (No matter how many variables it has) is between 0 and 2. Hence, theoretically, we show why NCF is stable since a random Boolean function has average sensitivity $\frac{n}{2}$. Finally we conjecture that the NCF attain the maximal average sensitivity if it has the maximal LAYER NUMBER $n-1$. Hence, we guess the uniform upper bound for the average sensitivity of any NCF can be reduced to 4/3 which is tight.
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