Yuan Li ; Frank Ingram ; Huaming Zhang - Certificate complexity and symmetry of nested canalizing functions

dmtcs:6191 - Discrete Mathematics & Theoretical Computer Science, November 26, 2021, vol. 23, no. 3 - https://doi.org/10.46298/dmtcs.6191
Certificate complexity and symmetry of nested canalizing functions

Authors: Yuan Li ; Frank Ingram ; Huaming Zhang

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.

Volume: vol. 23, no. 3
Section: Combinatorics
Published on: November 26, 2021
Accepted on: October 7, 2021
Submitted on: March 10, 2020
Keywords: Computer Science - Discrete Mathematics,Mathematics - Combinatorics,05A05, 05A15