Certificate complexity and symmetry of nested canalizing functions

Yuan Li,Huaming Zhang,Frank Ingram

doi:10.46298/dmtcs.6191

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.

Highlights

Nested canalizing functions (NCFs) were introduced in Kauffman et al (2003)
We investigate the relationship between the number of layers of an NCF and its number of symmetry levels
We obtained the formulas of the b-certificate complexity of any NCF for b = 0, 1

Summary

Introduction

Nested canalizing functions (NCFs) were introduced in Kauffman et al (2003) It was shown in Jarrah et al (2007) that they are identical to the unate cascade functions, which have been studied extensively in engineering and computer science. In Cook et al (1986), Cook et al introduced the notion of sensitivity as a combinatorial measure for Boolean functions. In Theorem 3.6 Li and Adeyeye (2019), the formula of the sensitivity of any NCF was obtained based on a characterization of NCFs from Theorem 4.2 Li et al (2013). It was shown that block sensitivity is the same as sensitivity for NCFs. In Moriznmi (2014), the author proved sensitivity is the same as the certificate complexity for readonce functions, a class of functions which include the NCFs, characterized as those that can be written using the logical conjunction, logical disjunction, and negation operations, where each variable appears at most once. We prove that there are more than n!2n−1 strongly asymmetric NCFs when n ≥ 4

Preliminaries

Certificate Complexity of NCFs

Symmetric Properties of NCFs

Conclusion