scholarly journals Certificate complexity and symmetry of nested canalizing functions

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.

scholarly journals Sensitivities and block sensitivities of elementary symmetric Boolean functions

2021 ◽  
Vol 15 (1) ◽  
pp. 434-453
Author(s):  
Jing Zhang ◽  
Yuan Li ◽  
John O. Adeyeye
Keyword(s):  
Information Technology ◽  
Computer Science ◽  
Boolean Functions ◽  
Regulatory Networks ◽  
Explicit Formulas ◽  
Complexity Measures ◽  
Symmetric Boolean Functions ◽  
Block Sensitivity

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.

Optimizations in computing the algebraic normal form transform of Boolean functions

2021 ◽  
Author(s):  
Maria Pashinska-Gadzheva ◽  
Valentin Bakoev ◽  
Iliya Bouyukliev ◽  
Dushan Bikov
Keyword(s):  
Normal Form ◽  
Boolean Functions ◽  
Algebraic Normal Form

scholarly journals Behavior of Analogical Inference w.r.t. Boolean Functions

2018 ◽  
Author(s):  
Miguel Couceiro ◽  
Nicolas Hug ◽  
Henri Prade ◽  
Gilles Richard
Keyword(s):  
Normal Form ◽  
Empirical Study ◽  
Boolean Functions ◽  
Hamming Distance ◽  
Affine Function ◽  
Analogical Inference ◽  
Wrong Prediction ◽  
Affine Functions ◽  
Classification Tasks ◽  
Algebraic Normal Form

It has been observed that a particular form of analogical inference, based on analogical proportions, yields competitive results in classification tasks. Using the algebraic normal form of Boolean functions, it has been shown that analogical prediction is always exact iff the labeling function is affine. We point out that affine functions are also meaningful when using another view of analogy. We address the accuracy of analogical inference for arbitrary Boolean functions and show that if a function is epsilon-close to an affine function, then the probability of making a wrong prediction is upper bounded by 4 epsilon. This result is confirmed by an empirical study showing that the upper bound is tight. It highlights the specificity of analogical inference, also characterized in terms of the Hamming distance.

On the Conjecture About the Linear Structures of Rotation Symmetric Boolean Functions

2017 ◽  
Vol 28 (07) ◽  
pp. 819-833
Author(s):  
Lei Sun ◽  
Fangwei Fu ◽  
Jian Liu
Keyword(s):  
Normal Form ◽  
Boolean Functions ◽  
Homogeneous Component ◽  
Linear Structures ◽  
Symmetric Boolean Functions ◽  
Variable Formula ◽  
Rotation Symmetric ◽  
Algebraic Normal Form ◽  
Rotation Symmetric Boolean Functions

In this paper, we study the conjecture that [Formula: see text]-variable ([Formula: see text] odd) rotation symmetric Boolean functions with degree [Formula: see text] have no non-zero linear structures. We show that if this class of RSBFs have non-zero linear structures, then the linear structures are invariant linear structures and the homogeneous component of degree [Formula: see text] in the function’s algebraic normal form has only two possibilities. Moreover, it is checked that the conjecture is true for [Formula: see text], and then a more explicit conjecture is proposed.

Chapter 13. Symmetry and Satisfiability

2021 ◽  
Author(s):  
Karem A. Sakallah
Keyword(s):  
Group Theory ◽  
Normal Form ◽  
Boolean Functions ◽  
Search Algorithm ◽  
Conjunctive Normal Form ◽  
Search Space ◽  
Decision Problems ◽  
Mathematical Language ◽  
Sat Solvers ◽  
Speed Up

Symmetry is at once a familiar concept (we recognize it when we see it!) and a profoundly deep mathematical subject. At its most basic, a symmetry is some transformation of an object that leaves the object (or some aspect of the object) unchanged. For example, a square can be transformed in eight different ways that leave it looking exactly the same: the identity “do-nothing” transformation, 3 rotations, and 4 mirror images (or reflections). In the context of decision problems, the presence of symmetries in a problem’s search space can frustrate the hunt for a solution by forcing a search algorithm to fruitlessly explore symmetric subspaces that do not contain solutions. Recognizing that such symmetries exist, we can direct a search algorithm to look for solutions only in non-symmetric parts of the search space. In many cases, this can lead to significant pruning of the search space and yield solutions to problems which are otherwise intractable. This chapter explores the symmetries of Boolean functions, particularly the symmetries of their conjunctive normal form (CNF) representations. Specifically, it examines what those symmetries are, how to model them using the mathematical language of group theory, how to derive them from a CNF formula, and how to utilize them to speed up CNF SAT solvers.

scholarly journals Convergence of the Chern–Moser–Beloshapka normal forms

2020 ◽  
Vol 2020 (765) ◽  
pp. 205-247
Author(s):  
Bernhard Lamel ◽  
Laurent Stolovitch
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Direct Proof ◽  
Sufficient Condition ◽  
Normalizing Transformation ◽  
General Sufficient Condition ◽  
Real Analytic ◽  
Moser Normal Form

AbstractIn this article, we give a normal form for real-analytic, Levi-nondegenerate submanifolds of{\mathbb{C}^{N}}of codimension{d\geq 1}under the action of formal biholomorphisms. We find a very general sufficient condition on the formal normal form that ensures that the normalizing transformation to this normal form is holomorphic. In the case{d=1}our methods in particular allow us to obtain a new and direct proof of the convergence of the Chern–Moser normal form.

Sign in / Sign up

Export Citation Format

Share Document