Symmetry Properties of Nested Canalyzing Functions

Many researchers have studied symmetry properties of various Boolean functions. A class of Boolean functions, called nested canalyzing functions (NCFs), has been used to model certain biological phenomena. We identify some interesting relationships between NCFs, symmetric Boolean functions and a generalization of symmetric Boolean functions, which we call $r$-symmetric functions (where $r$ is the symmetry level). Using a normalized representation for NCFs, we develop a characterization of when two variables of an NCF are symmetric. Using this characterization, we show that the symmetry level of an NCF $f$ can be easily computed given a standard representation of $f$. We also present an algorithm for testing whether a given $r$-symmetric function is an NCF. Further, we show that for any NCF $f$ with $n$ variables, the notion of strong asymmetry considered in the literature is equivalent to the property that $f$ is $n$-symmetric. We use this result to derive a closed form expression for the number of $n$-variable Boolean functions that are NCFs and strongly asymmetric. We also identify all the Boolean functions that are NCFs and symmetric.Comment: 17 page

Revealing the canalizing structure of Boolean functions: Algorithms and applications

Boolean functions can be represented in many ways including logical forms, truth tables, and polynomials. Additionally, Boolean functions have different canonical representations such as minimal disjunctive normal forms. Other canonical representation is based on the polynomial representation of Boolean functions where they can be written as a nested product of canalizing layers and a polynomial that contains the noncanalizing variables. In this paper we study the problem of identifying the canalizing layers format of Boolean functions. First, we show that the problem of finding the canalizing layers is NP-hard. Second, we present several algorithms for finding the canalizing layers of a Boolean function, discuss their complexities, and compare their performances. Third, we show applications where the computation of canalizing layers can be used for finding a disjunctive normal form of a nested canalizing function. Another application deals with the reverse engineering of Boolean networks with a prescribed layering format. Finally, implementations of our algorithms in Python and in the computer algebra system Macaulay2 are available at https://github.com/ckadelka/BooleanCanalization.Comment: 13 pages, 1 figur

Symmetry Properties of Nested Canalyzing Functions

Many researchers have studied symmetry properties of various Boolean functions. A class of Boolean functions, called nested canalyzing functions (NCFs), has been used to model certain biological phenomena. We identify some interesting relationships between NCFs, symmetric Boolean functions and a generalization of symmetric Boolean functions, which we call $r$-symmetric functions (where $r$ is the symmetry level). Using a normalized representation for NCFs, we develop a characterization of when two variables of an NCF are symmetric. Using this characterization, we show that the symmetry level of an NCF $f$ can be easily computed given a standard representation of $f$. We also present an algorithm for testing whether a given $r$-symmetric function is an NCF. Further, we show that for any NCF $f$ with $n$ variables, the notion of strong asymmetry considered in the literature is equivalent to the property that $f$ is $n$-symmetric. We use this result to derive a closed form expression for the number of $n$-variable Boolean functions that are NCFs and strongly asymmetric. We also identify all the Boolean functions that are NCFs and symmetric

Symmetry Properties of Nested Canalyzing Functions

Many researchers have studied symmetry properties of various Booleanfunctions. A class of Boolean functions, called nested canalyzing functions(NCFs), has been used to model certain biological phenomena. We identify someinteresting relationships between NCFs, symmetric Boolean functions and ageneralization of symmetric Boolean functions, which we call $r$-symmetricfunctions (where $r$ is the symmetry level). Using a normalized representationfor NCFs, we develop a characterization of when two variables of an NCF aresymmetric. Using this characterization, we show that the symmetry level of anNCF $f$ can be easily computed given a standard representation of $f$. We alsopresent an algorithm for testing whether a given $r$-symmetric function is anNCF. Further, we show that for any NCF $f$ with $n$ variables, the notion ofstrong asymmetry considered in the literature is equivalent to the propertythat $f$ is $n$-symmetric. We use this result to derive a closed formexpression for the number of $n$-variable Boolean functions that are NCFs andstrongly asymmetric. We also identify all the Boolean functions that are NCFsand symmetric.Comment: 17 page