Exact Value of the Nonmonotone Complexity of Boolean Functions
We study the complexity of the realization of Boolean functions by circuits in infinite complete bases containing all monotone functions with zero weight (cost of use) and finitely many nonmonotone functions with unit weight. The complexity of the realization of Boolean functions in the case where t...
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| Vydáno v: | Mathematical Notes Ročník 105; číslo 1-2; s. 28 - 35 |
|---|---|
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| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Moscow
Pleiades Publishing
01.01.2019
Springer Nature B.V |
| Témata: | |
| ISSN: | 0001-4346, 1067-9073, 1573-8876 |
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| Abstract | We study the complexity of the realization of Boolean functions by circuits in infinite complete bases containing all monotone functions with zero weight (cost of use) and finitely many nonmonotone functions with unit weight. The complexity of the realization of Boolean functions in the case where the only nonmonotone element of the basis is negation was completely described by A. A. Markov: the minimum number of negations sufficient for the realization of an arbitrary Boolean function
f
(the inversion complexity of the function
f
) is equal to ⌈log
2
(
d
(
f
) + 1)⌉, where
d
(
f
) is the maximum (over all increasing chains of sets of values of the variables) number of changes of the function value from 1 to 0. In the present paper, this result is generalized to the case of the computation of Boolean functions over an arbitrary basis
B
of prescribed form. It is shown that the minimum number of nonmonotone functions sufficient for computing an arbitrary Boolean function
f
is equal to ⌈log
2
(
d
(
f
)/
D
(
B
) +1)⌉, where
D
(
B
) = max
d
(
ω
); the maximum is taken over all nonmonotone functions ω of the basis
B
. |
|---|---|
| AbstractList | We study the complexity of the realization of Boolean functions by circuits in infinite complete bases containing all monotone functions with zero weight (cost of use) and finitely many nonmonotone functions with unit weight. The complexity of the realization of Boolean functions in the case where the only nonmonotone element of the basis is negation was completely described by A. A. Markov: the minimum number of negations sufficient for the realization of an arbitrary Boolean function f (the inversion complexity of the function f) is equal to ⌈log2(d(f) + 1)⌉, where d(f) is the maximum (over all increasing chains of sets of values of the variables) number of changes of the function value from 1 to 0. In the present paper, this result is generalized to the case of the computation of Boolean functions over an arbitrary basis B of prescribed form. It is shown that the minimum number of nonmonotone functions sufficient for computing an arbitrary Boolean function f is equal to ⌈log2(d(f)/D(B) +1)⌉, where D(B) = max d(ω); the maximum is taken over all nonmonotone functions ω of the basis B. We study the complexity of the realization of Boolean functions by circuits in infinite complete bases containing all monotone functions with zero weight (cost of use) and finitely many nonmonotone functions with unit weight. The complexity of the realization of Boolean functions in the case where the only nonmonotone element of the basis is negation was completely described by A. A. Markov: the minimum number of negations sufficient for the realization of an arbitrary Boolean function f (the inversion complexity of the function f ) is equal to ⌈log 2 ( d ( f ) + 1)⌉, where d ( f ) is the maximum (over all increasing chains of sets of values of the variables) number of changes of the function value from 1 to 0. In the present paper, this result is generalized to the case of the computation of Boolean functions over an arbitrary basis B of prescribed form. It is shown that the minimum number of nonmonotone functions sufficient for computing an arbitrary Boolean function f is equal to ⌈log 2 ( d ( f )/ D ( B ) +1)⌉, where D ( B ) = max d ( ω ); the maximum is taken over all nonmonotone functions ω of the basis B . |
| Author | Kochergin, V. V. Mikhailovich, A. V. |
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| Keywords | circuit complexity circuits of functional elements Boolean (logical) circuits inversion complexity nonmonotone complexity |
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| References | Markov (CR1) 1957; 116 Nechiporuk (CR6) 1962 Fischer (CR7) 1975 Kochergin, Mikhailovich (CR12) 2015 Blais, Canonne, Oliveira, Servedio, Tan (CR10) 2015; 40 CR9 Guo, Malkin, Oliveira, Rosen (CR11) 2015; 9014 Kochergin, Mikhailovich (CR13) 2016; 28 Markov (CR4) 1963; 150 Savage (CR3) 1976 Kochergin, Mikhailovich (CR14) 2015 Morizumi (CR8) 2008 Kochergin, Mikhailovich (CR15) 2015; 4 Lupanov (CR2) 1984 Gilbert (CR5) 1960 J. E. Savage (1038_CR3) 1976 1038_CR9 S. Guo (1038_CR11) 2015; 9014 M. J. Fischer (1038_CR7) 1975 H. Morizumi (1038_CR8) 2008 V. V. Kochergin (1038_CR15) 2015; 4 V. V. Kochergin (1038_CR13) 2016; 28 O. B. Lupanov (1038_CR2) 1984 V. V. Kochergin (1038_CR12) 2015 É I. I. Nechiporuk (1038_CR6) 1962 A. A. Markov (1038_CR1) 1957; 116 A. A. Markov (1038_CR4) 1963; 150 E. N. Gilbert (1038_CR5) 1960 E. Blais (1038_CR10) 2015; 40 V. V. Kochergin (1038_CR14) 2015 |
| References_xml | – year: 1975 ident: CR7 article-title: The complexity of negation–limited networks–a brief survey publication-title: Automata Theory and Formal Languages, Lecture Notes in Comput. Sci. – volume: 150 start-page: 477 issue: 3 year: 1963 end-page: 479 ident: CR4 article-title: On inversion complexity of a system of Boolean functions publication-title: Dokl. Akad. Nauk SSSR – volume: 28 start-page: 80 issue: 4 year: 2016 end-page: 90 ident: CR13 article-title: The minimum number of negations in circuits for systems of multi–valued functions publication-title: Diskret. Mat. doi: 10.4213/dm1394 – year: 2015 ident: CR14 publication-title: Some Extensions of the Inversion Complexity of Boolean Function – volume: 9014 start-page: 36 year: 2015 end-page: 65 ident: CR11 article-title: The power of negations in cryptography publication-title: Theory of Cryptography. Part I, Lecture Notes in Comput. Sci. doi: 10.1007/978-3-662-46494-6_3 – volume: 40 start-page: 512 year: 2015 ident: CR10 article-title: Learning circuits with few negations publication-title: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, LIPIcs. Leibniz Int. Proc. Inform. – year: 1976 ident: CR3 publication-title: The Complexity of Computing – year: 1984 ident: CR2 publication-title: Asymptotic Estimates of Complexity of Control Systems – year: 1960 ident: CR5 article-title: Theoretical–structural properties of closing switching functions publication-title: Collection of Papers in Cybernetics – ident: CR9 – year: 1962 ident: CR6 article-title: On complexity of circuits in some bases containing nontrivial elements with zero weights publication-title: Problems of Cybernetics – volume: 4 start-page: 24 issue: 30 year: 2015 end-page: 31 ident: CR15 article-title: On the complexity of circuits in bases containing monotone elements with zero weights publication-title: Prikl. Diskret. Mat. – year: 2008 ident: CR8 publication-title: A Note on the Inversion Complexity of Boolean Functions in Boolean Formulas – year: 2015 ident: CR12 publication-title: Inversion Complexity of Functions of Multi–Valued Logic – volume: 116 start-page: 917 issue: 6 year: 1957 end-page: 919 ident: CR1 article-title: On inversion complexity of systems of functions publication-title: Dokl. Akad. Nauk SSSR – volume-title: Inversion Complexity of Functions of Multi–Valued Logic year: 2015 ident: 1038_CR12 – volume-title: Problems of Cybernetics year: 1962 ident: 1038_CR6 – volume: 116 start-page: 917 issue: 6 year: 1957 ident: 1038_CR1 publication-title: Dokl. Akad. Nauk SSSR – ident: 1038_CR9 doi: 10.1007/978-3-642-24508-4 – volume: 40 start-page: 512 year: 2015 ident: 1038_CR10 publication-title: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, LIPIcs. Leibniz Int. Proc. Inform. – volume-title: Some Extensions of the Inversion Complexity of Boolean Function year: 2015 ident: 1038_CR14 – volume-title: The Complexity of Computing year: 1976 ident: 1038_CR3 – volume: 4 start-page: 24 issue: 30 year: 2015 ident: 1038_CR15 publication-title: Prikl. Diskret. Mat. – volume: 150 start-page: 477 issue: 3 year: 1963 ident: 1038_CR4 publication-title: Dokl. Akad. Nauk SSSR – volume-title: Asymptotic Estimates of Complexity of Control Systems year: 1984 ident: 1038_CR2 – volume-title: A Note on the Inversion Complexity of Boolean Functions in Boolean Formulas year: 2008 ident: 1038_CR8 – volume: 28 start-page: 80 issue: 4 year: 2016 ident: 1038_CR13 publication-title: Diskret. Mat. doi: 10.4213/dm1394 – volume-title: Automata Theory and Formal Languages, Lecture Notes in Comput. Sci. year: 1975 ident: 1038_CR7 – volume-title: Collection of Papers in Cybernetics year: 1960 ident: 1038_CR5 – volume: 9014 start-page: 36 year: 2015 ident: 1038_CR11 publication-title: Theory of Cryptography. Part I, Lecture Notes in Comput. Sci. doi: 10.1007/978-3-662-46494-6_3 |
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| SubjectTerms | Boolean algebra Boolean functions Codes Complexity Markov processes Mathematics Mathematics and Statistics Monotone functions Weight |
| Title | Exact Value of the Nonmonotone Complexity of Boolean Functions |
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