嵌套渠化函数的证书复杂性和对称性
Certificate complexity and symmetry of nested canalizing functions
摘要：
布尔嵌套渠化函数在分子调控网络、工程和计算机科学中有着重要的应用。在本文中，我们研究他们的证书复杂性。对于这两个布尔值$b\in\{0,1\}$，我们得到了一个公式的$b$ -证书的复杂性，因此，我们开发了一个直接证明的证书复杂性公式的NCF。对称性是布尔函数的另一个有趣的性质，我们显着简化了最近的一些定理的证明NCF的部分对称性。我们还描述了$s$ -对称NCF的代数正规形。得到了$s=1,\dots,n$的$n$ -变量$s$ -对称布尔NCF集的基数的一般公式.特别是，我们列举了强非对称布尔NCF。
英文摘要：
Boolean nested canalizing functions (NCFs) have important applications in molecular regulatory networks, engineering and computer science. In this paper, we study their certificate complexity. For both Boolean values $b\in\{0,1\}$, we obtain a formula for $b$-certificate complexity and consequently, we develop a direct proof of the certificate complexity formula of an NCF. Symmetry is another interesting property of Boolean functions and we significantly simplify the proofs of some recent theorems about partial symmetry of NCFs. We also describe the algebraic normal form of $s$-symmetric NCFs. We obtain the general formula of the cardinality of the set of $n$-variable $s$-symmetric Boolean NCFs for $s=1,\dots,n$. In particular, we enumerate the strongly asymmetric Boolean NCFs.
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