Zhegalkin Polynomial of a Multiary Sole Sufficient Operator

— Sole sufficient operators are of particular interest among functionally complete sets of Boolean functions. They have a wide range of applicability and are not limited to the binarity case. In this paper, we formulate conditions imposed on the Zhegalkin polynomial coefficients that are necessary a...

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Vydané v:Automatic control and computer sciences Ročník 58; číslo 7; s. 808 - 826
Hlavní autori: Bystrov, L. Y., Kuzmin, E. V.
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Moscow Pleiades Publishing 01.12.2024
Springer Nature B.V
Predmet:
Boolean
Boolean functions
Codes
Computer Science
Control Structures and Microprogramming
Operators (mathematics)
Polynomials
Variables
dual remainder polynomial
constant-preserving Boolean functions
self-dual Boolean functions
Zhegalkin polynomial
leading coefficient
sole sufficient operator
Sheffer function
precomplete classes
ISSN:0146-4116, 1558-108X
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Abstract — Sole sufficient operators are of particular interest among functionally complete sets of Boolean functions. They have a wide range of applicability and are not limited to the binarity case. In this paper, we formulate conditions imposed on the Zhegalkin polynomial coefficients that are necessary and sufficient for the polynomial to correspond to a sole sufficient operator. The polynomial representation of constant-preserving Boolean functions is considered. It is shown that the properties of monotonicity and linearity do not need to be specifically considered when describing a sole sufficient operator. The concept of a dual remainder polynomial is introduced; the value of it allows one to determine the self-duality of a Boolean function. It is proven that a Boolean function preserving 0 and 1 or preserving neither 0 nor 1 is self-dual if and only if the dual remainder of the corresponding Zhegalkin polynomial is 0 for any sets of values of the function variables. The system of leading coefficients is obtained based on this fact. The solution of the system makes it possible to formulate a criterion for the self-duality of a Boolean function represented by a Zhegalkin polynomial, which imposes necessary and sufficient conditions on the polynomial coefficients. Thus, it is shown that Zhegalkin polynomials are a rather convenient tool for studying precomplete classes of Boolean functions.
AbstractList Abstract—Sole sufficient operators are of particular interest among functionally complete sets of Boolean functions. They have a wide range of applicability and are not limited to the binarity case. In this paper, we formulate conditions imposed on the Zhegalkin polynomial coefficients that are necessary and sufficient for the polynomial to correspond to a sole sufficient operator. The polynomial representation of constant-preserving Boolean functions is considered. It is shown that the properties of monotonicity and linearity do not need to be specifically considered when describing a sole sufficient operator. The concept of a dual remainder polynomial is introduced; the value of it allows one to determine the self-duality of a Boolean function. It is proven that a Boolean function preserving 0 and 1 or preserving neither 0 nor 1 is self-dual if and only if the dual remainder of the corresponding Zhegalkin polynomial is 0 for any sets of values of the function variables. The system of leading coefficients is obtained based on this fact. The solution of the system makes it possible to formulate a criterion for the self-duality of a Boolean function represented by a Zhegalkin polynomial, which imposes necessary and sufficient conditions on the polynomial coefficients. Thus, it is shown that Zhegalkin polynomials are a rather convenient tool for studying precomplete classes of Boolean functions.
— Sole sufficient operators are of particular interest among functionally complete sets of Boolean functions. They have a wide range of applicability and are not limited to the binarity case. In this paper, we formulate conditions imposed on the Zhegalkin polynomial coefficients that are necessary and sufficient for the polynomial to correspond to a sole sufficient operator. The polynomial representation of constant-preserving Boolean functions is considered. It is shown that the properties of monotonicity and linearity do not need to be specifically considered when describing a sole sufficient operator. The concept of a dual remainder polynomial is introduced; the value of it allows one to determine the self-duality of a Boolean function. It is proven that a Boolean function preserving 0 and 1 or preserving neither 0 nor 1 is self-dual if and only if the dual remainder of the corresponding Zhegalkin polynomial is 0 for any sets of values of the function variables. The system of leading coefficients is obtained based on this fact. The solution of the system makes it possible to formulate a criterion for the self-duality of a Boolean function represented by a Zhegalkin polynomial, which imposes necessary and sufficient conditions on the polynomial coefficients. Thus, it is shown that Zhegalkin polynomials are a rather convenient tool for studying precomplete classes of Boolean functions.
Author Bystrov, L. Y.
Kuzmin, E. V.
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  orcidid: 0000-0002-0610-5466
  surname: Bystrov
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  organization: Demidov Yaroslavl State University (YSU)
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  givenname: E. V.
  orcidid: 0000-0003-0500-306X
  surname: Kuzmin
  fullname: Kuzmin, E. V.
  email: kuzmin@uniyar.ac.ru
  organization: Demidov Yaroslavl State University (YSU)
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Copyright Allerton Press, Inc. 2024 ISSN 0146-4116, Automatic Control and Computer Sciences, 2024, Vol. 58, No. 7, pp. 808–826. © Allerton Press, Inc., 2024.Russian Text © The Author(s), 2023, published in Modelirovanie i Analiz Informatsionnykh Sistem, 2023, Vol. 30, No. 2, pp. 106–127.
Allerton Press, Inc. 2024.
Copyright_xml – notice: Allerton Press, Inc. 2024 ISSN 0146-4116, Automatic Control and Computer Sciences, 2024, Vol. 58, No. 7, pp. 808–826. © Allerton Press, Inc., 2024.Russian Text © The Author(s), 2023, published in Modelirovanie i Analiz Informatsionnykh Sistem, 2023, Vol. 30, No. 2, pp. 106–127.
– notice: Allerton Press, Inc. 2024.
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constant-preserving Boolean functions
self-dual Boolean functions
Zhegalkin polynomial
leading coefficient
sole sufficient operator
Sheffer function
precomplete classes
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Abstract—Sole sufficient operators are of particular interest among functionally complete sets of Boolean functions. They have a wide range of applicability...
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SubjectTerms Boolean
Boolean functions
Codes
Computer Science
Control Structures and Microprogramming
Operators (mathematics)
Polynomials
Variables
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