The complexity of Boolean functions in the Reed–Muller polynomials class

This paper considers the problem of transforametion of Boolean functions into canonical polarized polynomials (Reed–Muller polynomials). Two Shannon functions are introduced to estimate the complexity of Boolean functions in the polynomials class under consideration. We propose three Boolean functio...

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Vydané v:	Automatic control and computer sciences Ročník 51; číslo 5; s. 285 - 293
Hlavný autor:	Suprun, V. P.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Moscow Pleiades Publishing 01.09.2017 Springer Nature B.V
Predmet:	Boolean algebra Boolean functions Codes Complexity Computer Science Control Structures and Microprogramming Functions (mathematics) Mathematical analysis Polynomials polynomial complexity Reed–Muller polynomial Zhegalkin polynomial symmetric Boolean function triangle method logical scheme Boolean function Shannon functions Boolean derivative
ISSN:	0146-4116, 1558-108X
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Abstract	This paper considers the problem of transforametion of Boolean functions into canonical polarized polynomials (Reed–Muller polynomials). Two Shannon functions are introduced to estimate the complexity of Boolean functions in the polynomials class under consideration. We propose three Boolean functions of n variables whose complexity (in terms of the number of terms) coincides with value. We investigate the properties of functions and propose their schematic realization on elements AND, XOR, and NAND.
AbstractList	This paper considers the problem of transforametion of Boolean functions into canonical polarized polynomials (Reed–Muller polynomials). Two Shannon functions are introduced to estimate the complexity of Boolean functions in the polynomials class under consideration. We propose three Boolean functions of n variables whose complexity (in terms of the number of terms) coincides with value. We investigate the properties of functions and propose their schematic realization on elements AND, XOR, and NAND. This paper considers the problem of transforametion of Boolean functions into canonical polarized polynomials (Reed–Muller polynomials). Two Shannon functions are introduced to estimate the complexity of Boolean functions in the polynomials class under consideration. We propose three Boolean functions of n variables whose complexity (in terms of the number of terms) coincides with value. We investigate the properties of functions and propose their schematic realization on elements AND, XOR, and NAND.
Author	Suprun, V. P.
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Keywords	polynomial complexity Reed–Muller polynomial Zhegalkin polynomial symmetric Boolean function triangle method logical scheme Boolean function Shannon functions Boolean derivative
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References	SuprunV.P.KorobkoF.S.Synthesis of logical devices for calculating self-dual symmetric Boolean functionsAvtom. Vychisl. Tekh.201412635 SuprunV.P.Complexity of Boolean functions in the class of canonical polarized polynomialsDiskretn. Mat.19935211111512509610829.94013 PospelovD.A.Logicheskie metody analiza i sinteza skhem1974MoscowEnergiya SuprunV.P.GorodetskiiD.A.The matrix method for polynomial expansion of symmetric boolean functionsAvtom. Vychisl. Tekh.20131512 PeryazevN.A.Complexity of Boolean functions in the class of polynomial polarized formsAlgebra Logika1995332332613644700863.94034 SuprunV.P.Table method of polynomial expansion of Boolean functionsKibernetika198711161178926640671.94021 SuprunV.P.Estimations of Shennon’s function for polarity Reed–Muller expressionsProc. of the IFIP WG 10.5 Workshop on Applications of the Reed–Muller Expansion in Circuit Design1993107114 Avgul’L.B.SuprunV.P.Synthesis of high-speed logic circuits by the cascade methodIzv. Vuzov. Priborostr.199333136 V.P. Suprun (6598_CR5) 1993 L.B. Avgul (6598_CR8) 1993; 3 V.P. Suprun (6598_CR2) 2013; 1 V.P. Suprun (6598_CR3) 1993; 5 N.A. Peryazev (6598_CR4) 1995; 3 V.P. Suprun (6598_CR7) 2014; 1 V.P. Suprun (6598_CR6) 1987; 1 D.A. Pospelov (6598_CR1) 1974
References_xml	– reference: SuprunV.P.Complexity of Boolean functions in the class of canonical polarized polynomialsDiskretn. Mat.19935211111512509610829.94013 – reference: PeryazevN.A.Complexity of Boolean functions in the class of polynomial polarized formsAlgebra Logika1995332332613644700863.94034 – reference: PospelovD.A.Logicheskie metody analiza i sinteza skhem1974MoscowEnergiya – reference: SuprunV.P.Estimations of Shennon’s function for polarity Reed–Muller expressionsProc. of the IFIP WG 10.5 Workshop on Applications of the Reed–Muller Expansion in Circuit Design1993107114 – reference: SuprunV.P.KorobkoF.S.Synthesis of logical devices for calculating self-dual symmetric Boolean functionsAvtom. Vychisl. Tekh.201412635 – reference: SuprunV.P.GorodetskiiD.A.The matrix method for polynomial expansion of symmetric boolean functionsAvtom. Vychisl. Tekh.20131512 – reference: Avgul’L.B.SuprunV.P.Synthesis of high-speed logic circuits by the cascade methodIzv. Vuzov. Priborostr.199333136 – reference: SuprunV.P.Table method of polynomial expansion of Boolean functionsKibernetika198711161178926640671.94021 – volume: 3 start-page: 323 year: 1995 ident: 6598_CR4 publication-title: Algebra Logika – volume: 5 start-page: 111 issue: 2 year: 1993 ident: 6598_CR3 publication-title: Diskretn. Mat. – volume: 3 start-page: 31 year: 1993 ident: 6598_CR8 publication-title: Izv. Vuzov. Priborostr. – volume: 1 start-page: 116 year: 1987 ident: 6598_CR6 publication-title: Kibernetika – start-page: 107 volume-title: Proc. of the IFIP WG 10.5 Workshop on Applications of the Reed–Muller Expansion in Circuit Design year: 1993 ident: 6598_CR5 – volume: 1 start-page: 5 year: 2013 ident: 6598_CR2 publication-title: Avtom. Vychisl. Tekh. – volume-title: Logicheskie metody analiza i sinteza skhem year: 1974 ident: 6598_CR1 – volume: 1 start-page: 26 year: 2014 ident: 6598_CR7 publication-title: Avtom. Vychisl. Tekh.
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SubjectTerms	Boolean algebra Boolean functions Codes Complexity Computer Science Control Structures and Microprogramming Functions (mathematics) Mathematical analysis Polynomials
Title	The complexity of Boolean functions in the Reed–Muller polynomials class
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