scholarly journals An Efficient Algorithm for the Calculation of Generalized Adding and Arithmetic Transforms From Disjoint Cubes of Boolean Functions

VLSI Design ◽  
1999 ◽  
Vol 9 (2) ◽  
pp. 135-146 ◽  
Author(s):  
Bogdan J. Falkowski ◽  
Chip-Hong Chang
Keyword(s):  
Boolean Functions ◽  
Efficient Algorithm ◽  
Truth Table ◽  
Computer Memory ◽  
Reduced Representation ◽  
Spectral Coefficients

A new algorithm is given that converts a reduced representation of Boolean functions in the form of disjoint cubes to Generalized Adding and Arithmetic spectra. Since the known algorithms that generate Adding and Arithmetic spectra always start from the truth table of Boolean functions the method presented computes faster with a smaller computer memory. The method is extremely efficient for such Boolean functions that are described by only few disjoint cubes and it allows the calculation of only selected spectral coefficients, or all the coefficients can be calculated in parallel.

IDENTIFICATION AND REALIZATION OF LINEARLY SEPARABLE BOOLEAN FUNCTIONS VIA CELLULAR NEURAL NETWORKS

2008 ◽  
Vol 18 (11) ◽  
pp. 3299-3308 ◽  
Author(s):  
BO MI ◽  
XIAOFENG LIAO ◽  
CHUANDONG LI
Keyword(s):  
Neural Networks ◽  
Directed Graph ◽  
Boolean Functions ◽  
Cellular Neural Networks ◽  
Truth Table

In this paper, an effective method for identifying and realizing linearly separable Boolean functions (LSBF) of six variables via Cellular Neural Networks (CNN) is presented. We characterized the basic relations between CNN genes and the truth table of Boolean functions. In order to implement LSBF independently, a directed graph is employed to sort the offset levels according to the truth table. Because any linearly separable Boolean gene (LSBG) can be derived separately, our method will be more practical than former schemes [Chen & Chen, 2005a, 2005b; Chen & He, 2006].

Number of spectral coefficients necessary to identify a class of Boolean functions

1982 ◽  
Vol 18 (13) ◽  
pp. 577 ◽  
Author(s):  
J. Muzio ◽  
D.M. Miller ◽  
S.L. Hurst
Keyword(s):  
Boolean Functions ◽  
Spectral Coefficients

scholarly journals Developing the minimization of a polynomial normal form of boolean functions by the method of figurative transformations

2021 ◽  
Vol 2 (4 (110)) ◽  
pp. 22-37
Author(s):  
Mykhailo Solomko ◽  
Iuliia Batyshkina ◽  
Nataliia Khomiuk ◽  
Yakiv Ivashchuk ◽  
Natalia Shevtsova
Keyword(s):  
Experimental Study ◽  
Normal Form ◽  
Boolean Functions ◽  
Transformation Method ◽  
Truth Table ◽  
Basis Functions ◽  
Polynomial Basis ◽  
Polynomial Functions ◽  
Sufficient Resource ◽  
Polynomial Normal Form

This paper reports a study that has established the possibility of improving the effectiveness of the method of figurative transformations in order to minimize Boolean functions on the Reed-Muller basis. Such potential prospects in the analytical method have been identified as a sequence in the procedure of inserting the same conjuncterms of polynomial functions followed by the operation of super-gluing the variables. The extension of the method of figurative transformations to the process of simplifying the functions of the polynomial basis involved the developed algebra in terms of the rules for simplifying functions in the Reed-Muller basis. It was established that the simplification of Boolean functions of the polynomial basis by a figurative transformation method is based on a flowchart with repetition, which is actually the truth table of the predefined function. This is a sufficient resource to minimize functions that makes it possible not to refer to such auxiliary objects as Karnaugh maps, Weich charts, cubes, etc. A perfect normal form of the polynomial basis functions can be represented by binary sets or a matrix that would represent the terms of the functions and the addition operation by module two for them. The experimental study has confirmed that the method of figurative transformations that employs the systems of 2-(n, b)-design, and 2-(n, x/b)-design in the first matrix improves the efficiency of minimizing Boolean functions. That also simplifies the procedure for finding a minimum function on the Reed-Muller basis. Compared to analogs, this makes it possible to enhance the performance of minimizing Boolean functions by 100‒200 %. There is reason to assert the possibility of improving the efficiency of minimizing Boolean functions in the Reed-Muller basis by a method of figurative transformations. This is ensured by using more complex algorithms to simplify logical expressions involving a procedure of inserting the same function terms in the Reed-Muller basis, followed by the operation of super-gluing the variables.

scholarly journals An improved spectral classification of Boolean functions based on an extended set of invariant operations

2018 ◽  
Vol 31 (2) ◽  
pp. 189-205
Author(s):  
Milena Stankovic ◽  
Claudio Moraga ◽  
Radomir Stankovic
Keyword(s):  
Boolean Functions ◽  
Quadratic Forms ◽  
Bent Functions ◽  
Engineering Practice ◽  
Theoretical Point ◽  
Spectral Coefficients ◽  
Invariant Operation ◽  
Digital System Design ◽  
Spectral Invariant

Boolean functions expressing some particular properties often appear in engineering practice. Therefore, a lot of research efforts are put into exploring different approaches towards classification of Boolean functions with respect to various criteria that are typically selected to serve some specific needs of the intended applications. A classification is considered to be strong if there is a reasonably small number of different classes for a given number of variables n and it it desir able that classification rules are simple. A classification with respect to Walsh spectral coefficients, introduced formerly for digital system design purposes, appears to be useful in the context of Boolean functions used in cryptography, since it is in a way compatible with characterization of cryptographically interesting functions through Walsh spectral coefficients. This classification is performed in terms of certain spectral invariant operations. We show by introducing a new spectral invariant operation in the Walsh domain, that by starting from n?5, some classes of Boolean functions can be merged which makes the classification stronger, and from the theoretical point of view resolves a problem raised already in seventies of the last century. Further, this new spectral invariant operation can be used in constructing bent functions from bent functions represented by quadratic forms.

Improved asymptotic estimates for the numbers of correlation-immune and k-resilient vectorial Boolean functions

2019 ◽  
Vol 29 (3) ◽  
pp. 195-213 ◽  
Author(s):  
Konstantin N. Pankov
Keyword(s):  
Boolean Functions ◽  
Limit Theorems ◽  
Local Limit ◽  
Weight Vector ◽  
Asymptotic Estimates ◽  
Linear Combinations ◽  
Vectorial Boolean Functions ◽  
Spectral Coefficients ◽  
Coordinate Functions ◽  
Local Limit Theorems

Abstract We refine local limit theorems for the distribution of a part of the weight vector of subfunctions and for the distribution of a part of the vector of spectral coefficients of linear combinations of coordinate functions of a random binary mapping. These theorems are used to derive improved asymptotic estimates for the numbers of correlation-immune and k-resilient vectorial Boolean functions.

The Iota-Delta Function as an Alternative to Boolean Formalism

2018 ◽  
Vol 29 (03) ◽  
pp. 415-423
Author(s):  
Luan Carlos de Sena Monteiro Ozelim ◽  
Andre Luis Brasil Cavalcante
Keyword(s):  
Boolean Functions ◽  
Delta Function ◽  
Truth Table ◽  
Easy Task ◽  
Real Polynomials ◽  
Truth Tables

Representing binary truth tables is an easy task to Boolean functions. When the number of Boolean variables increases, the complexity of the Boolean functions also increases. Thus, alternative ways of representing binary truth tables are necessary. It is known that this can be accomplished by means of other approaches, such as real polynomials. In the present paper it is shown that the recently introduced iota-delta function can be used to represent every binary truth table.

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