Sensitivities and block sensitivities of elementary symmetric Boolean functions

Abstract Boolean functions have important applications in molecular regulatory networks, engineering, cryptography, information technology, and computer science. Symmetric Boolean functions have received a lot of attention in several decades. Sensitivity and block sensitivity are important complexity measures of Boolean functions. In this paper, we study the sensitivity of elementary symmetric Boolean functions and obtain many explicit formulas. We also obtain a formula for the block sensitivity of symmetric Boolean functions and discuss its applications in elementary symmetric Boolean functions.

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Minimum complexity drives regulatory logic in Boolean models of living systems

10.1101/2021.09.20.461164 ◽

2021 ◽

Author(s):

Ajay Subbaroyan ◽

Olivier C. Martin ◽

Areejit Samal

Keyword(s):

Biological Networks ◽

Boolean Functions ◽

Regulatory Networks ◽

Statistical Physics ◽

Boolean Networks ◽

Average Sensitivity ◽

Boolean Models ◽

Complexity Measures ◽

Random Boolean Networks ◽

Update Rules

The properties of random Boolean networks as models of gene regulation have been investigated extensively by the statistical physics community. In the past two decades, there has been a dramatic increase in the reconstruction and analysis of Boolean models of biological networks. In such models, neither network topology nor Boolean functions (or logical update rules) should be expected to be random. In this contribution, we focus on biologically meaningful types of Boolean functions, and perform a systematic study of their preponderance in gene regulatory networks. By applying the k[P] classification based on number of inputs k and bias P of functions, we find that most Boolean functions astonishingly have odd bias in a reference biological dataset of 2687 functions compiled from published models. Subsequently, we are able to explain this observation along with the enrichment of read-once functions (RoFs) and its subset, nested canalyzing functions (NCFs), in the reference dataset in terms of two complexity measures: Boolean complexity based on string lengths in formal logic which is yet unexplored in the biological context, and the average sensitivity. Minimizing the Boolean complexity naturally sifts out a subset of odd-biased Boolean functions which happen to be the RoFs. Finally, we provide an analytical proof that NCFs minimize not only the Boolean complexity, but also the average sensitivity in their k[P] set.

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Certificate complexity and symmetry of nested canalizing functions

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.6191 ◽

2021 ◽

Vol vol. 23, no. 3 (Combinatorics) ◽

Author(s):

Yuan Li ◽

Frank Ingram ◽

Huaming Zhang

Keyword(s):

Normal Form ◽

Computer Science ◽

General Formula ◽

Boolean Functions ◽

Regulatory Networks ◽

Direct Proof ◽

Interesting Property ◽

Partial Symmetry ◽

Algebraic Normal Form

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.

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