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 gen...

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Bibliographic Details
Published inDiscrete Mathematics and Theoretical Computer Science Vol. 21; no. 4; pp. 1 - 17
Main Authors Rosenkrantz, Daniel J, Marathe, Madhav V, Ravi, S.S, Stearns, Richard E
Format Journal Article
LanguageEnglish
Published Nancy DMTCS 01.08.2019
Subjects
Algorithms
Asymmetry
Boolean
Boolean algebra
Boolean functions
Codes
Mathematical research
Representations
Symmetry
United States
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Summary: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.
ISSN:1462-7264
1365-8050