Certificate complexity and symmetry of nested canalizing functions

TL;DR

This paper analyzes the complexity and symmetry properties of nested canalizing functions, providing formulas for certificate complexity, symmetry classifications, and enumeration of specific subclasses, with implications for computational and biological systems.

Contribution

It introduces explicit formulas for certificate complexity of NCFs and simplifies proofs related to their symmetry properties, expanding understanding of their algebraic structure.

Findings

Derived formulas for b-certificate complexity of NCFs

Simplified proofs of partial symmetry properties of NCFs

Enumerated classes of s-symmetric and strongly asymmetric NCFs

Abstract

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.