scholarly journals Properties of Symmetric Boolean functions

2016 ◽  
Vol 12 (1) ◽  
pp. 5-24
Author(s):  
L. Haviarová ◽  
E. Toman
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Normal Forms ◽  
Special Property ◽  
Interval Graphs ◽  
Vertex Degree ◽  
Symmetric Boolean Function ◽  
Symmetric Boolean Functions

Abstract In the present paper we consider symmetric Boolean functions with special property. We study properties of the maximal intervals of these functions. Later we show characteristics of corresponding interval graphs and simplified interval graphs. Specifically we prove, that these two graphs are isomorphic for symmetric Boolean function. Then we obtain the vertex degree of these graphs. We discuss also disjunctive normal forms.

scholarly journals The Weight and Nonlinearity of 2-rotation Symmetric Cubic Boolean Function

2015 ◽  
Vol 7 (2) ◽  
pp. 187 ◽  
Author(s):  
Hongli Liu
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Symmetric Functions ◽  
Recursive Formula ◽  
Symmetric Boolean Function ◽  
Symmetric Boolean Functions ◽  
Rotation Symmetric ◽  
Rotation Symmetric Boolean Function ◽  
Rotation Symmetric Boolean Functions

The conceptions of $\chi$-value and K-rotation symmetric Boolean functions are introduced by Cusick. K-rotation symmetric Boolean functions are a special rotation symmetric functions, which are invariant under the $k-th$ power of $\rho$.In this paper, we discuss cubic 2-value 2-rotation symmetric Boolean function with $2n$ variables, which denoted by $F^{2n}(x^{2n})$. We give the recursive formula of weight of $F^{2n}(x^{2n})$, and prove that the weight of $F^{2n}(x^{2n})$ is the same as its nonlinearity.

ON THE NONEXISTENCE of BENT FUNCTIONS

2011 ◽  
Vol 22 (06) ◽  
pp. 1431-1438 ◽  
Author(s):  
YIN ZHANG ◽  
MEICHENG LIU ◽  
DONGDAI LIN
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Normal Forms ◽  
Bent Functions

In this paper, we study the nonexistence of bent functions in the class of Boolean functions without monomials of degree less than d in their algebraic normal forms (ANF). We prove that n-variable Boolean functions in such class are not bent when there are not more than n + d - 3 monomials in their ANFs. We also show that an n-variable Boolean function is not bent if it has no monomial of degree less than ⌈3n/8 + 3/4⌉ in its ANF.

scholarly journals Periodic Fourier representation of Boolean functions

2019 ◽  
Vol 19 (5&6) ◽  
pp. 392-412
Author(s):  
Ryuhei Mori
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Bell Inequalities ◽  
Polynomial Representation ◽  
Symmetric Boolean Function ◽  
Fourier Representation ◽  
Minimum Number ◽  
New Type ◽  
Adaptive Measurement ◽  
Small Periodic

In this work, we consider a new type of Fourier-like representation of Boolean function f\colon\{+1,-1\}^n\to\{+1,-1\}: f(x) = \cos(\pi\sum_{S\subseteq[n]}\phi_S \prod_{i\in S} x_i). This representation, which we call the periodic Fourier representation, of Boolean function is closely related to a certain type of multipartite Bell inequalities and non-adaptive measurement-based quantum computation with linear side-processing NMQCp. The minimum number of non-zero coefficients in the above representation, which we call the periodic Fourier sparsity, is equal to the required number of qubits for the exact computation of f by \NMQCp. Periodic Fourier representations are not unique, and can be directly obtained both from the Fourier representation and the F_2-polynomial representation. In this work, we first show that Boolean functions related to ZZZZ-polynomial have small periodic Fourier sparsities. Second, we show that the periodic Fourier sparsity is at least 2^{\deg_{\mathbb{F}_2}(f)}-1, which means that NMQCp efficiently computes a Boolean function $f$ if and only if F_2-degree of f is small. Furthermore, we show that any symmetric Boolean function, e.g., AND_n, Mod^3_n, Maj_n, etc, can be exactly computed by depth-2 NMQCp using a polynomial number of qubits, that implies exponential gaps between NMQCp and depth-2 NMQCp.

scholarly journals Linear Recurrences and Asymptotic Behavior of Exponential Sums of Symmetric Boolean Functions

10.37236/2004 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Francis N. Castro ◽  
Luis A. Medina
Keyword(s):  
Asymptotic Behavior ◽  
Boolean Functions ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Linear Recurrence ◽  
Symmetric Boolean Function ◽  
Linear Recurrences ◽  
Symmetric Boolean Functions ◽  
Integer Coefficients ◽  
Homogeneous Linear

In this paper we give an improvement of the degree of the homogeneous linear recurrence with integer coefficients that exponential sums of symmetric Boolean functions satisfy. This improvement is tight. We also compute the asymptotic behavior of symmetric Boolean functions and provide a formula that allows us to determine if a symmetric boolean function is asymptotically not balanced. In particular, when the degree of the symmetric function is a power of two, then the exponential sum is much smaller than $2^n$.

scholarly journals The Stochastic Boolean Function Evaluation problem for symmetric Boolean functions

2022 ◽  
Vol 309 ◽  
pp. 269-277
Author(s):  
Dimitrios Gkenosis ◽  
Nathaniel Grammel ◽  
Lisa Hellerstein ◽  
Devorah Kletenik
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Function Evaluation ◽  
Symmetric Boolean Functions ◽  
Evaluation Problem

Exact Quantum 1-Query Algorithms and Complexity

SPIN ◽  
2021 ◽  
pp. 2140001
Author(s):  
Daowen Qiu ◽  
Guoliang Xu
Keyword(s):  
Boolean Function ◽  
Decision Trees ◽  
Future Study ◽  
Boolean Functions ◽  
Query Complexity ◽  
Symmetric Boolean Functions ◽  
Exact Quantum ◽  
Algorithms And Complexity ◽  
Special Case ◽  
Query Algorithms

Deutsch–Jozsa problem (D–J) has exact quantum 1-query complexity (“exact” means no error), but requires super-exponential queries for the optimal classical deterministic decision trees. D–J problem is equivalent to a symmetric partial Boolean function, and in fact, all symmetric partial Boolean functions having exact quantum 1-query complexity have been found out and these functions can be computed by D–J algorithm. A special case is that all symmetric Boolean functions with exact quantum 1-query complexity follow directly and these functions are also all total Boolean functions with exact quantum 1-query complexity obviously. Then there are pending problems concerning partial Boolean functions having exact quantum 1-query complexity and new results have been found, but some problems are still open. In this paper, we review these results regarding exact quantum 1-query complexity and in particular, we also obtain a new result that a partial Boolean function with exact quantum 1-query complexity is constructed and it cannot be computed by D–J algorithm. Further problems are pointed out for future study.

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