The generalized complexity of linear Boolean functions
-
Nikolay P. Redkin
Abstract
We study generalized (in terms of bases) complexity of implementation of linear Boolean functions by Boolean circuits in arbitrary functionally complete bases; the complexity of a circuit is defined as the number of gates. Let L*(n) be the minimal number of gates sufficient for implementation of an arbitrary linear Boolean function of n variables in an arbitrary functionally complete basis. We show that L*(0) = L*(1) = 3 and L*(n) = 7(n – 1) for any natural n ≥ 2.
Originally published in Diskretnaya Matematika (2018) 30, №4, 89–96 (in Russian).
Funding
Research was supported by Russian Foundation for Basic Research (project 18.01.00337 “Problems of synthesis, complexity and reliability in theory of control systems”).
References
[1] Lupanov O. B., Asymptotic complexity estimates of control systems, MSU Publ., Moscow, 1984 (in Russian), 138 pp.Suche in Google Scholar
[2] Redkin N. P., “A generalization of Shannon function”, Discrete Math. Appl., 28:5 (2018), 309–318.10.1515/dma-2018-0027Suche in Google Scholar
[3] Yablonsky S.V., Introduction to Discrete Mathematics, Vysshaya shkola, Moscow, 2003 (in Russian), 384 pp.Suche in Google Scholar
[4] Redkin N. P., “The proof of minimality of some circuits of functional elements”, Problemy kibernetiki, 1970,№23, 83–101 (in Russian).Suche in Google Scholar
[5] Redkin N. P., “On the minimal implementation of a linear function by a circuit of functional elements”, Kibernetika, 1971, №6, 31.38 (in Russian).10.1007/BF01068820Suche in Google Scholar
[6] Shkrebela I. S., “On complexity of realisation of linear Boolean functions by circuits of functional elements over the basis {x → y, x}”, Discrete Math. Appl., 13:5 (2003), 483–496.10.1163/156939203322694754Suche in Google Scholar
[7] Kombarov Yu. A., “The minimal circuits for linear Boolean functions”, Moscow University Mechanics Bulletin, 66:6 (2011), 260–263.10.3103/S0027132211060076Suche in Google Scholar
[8] Kombarov Yu. A., “On minimal implementations of linear boolean functions”, J. Appl. Industr. Math., 19:3 (2012), 39–57.Suche in Google Scholar
[9] Kombarov Yu. A., “On minimal circuits for linear functions over some bases”, Discrete Math. Appl., 23:1 (2013), 39–51.10.1515/dma-2013-002Suche in Google Scholar
[10] Kombarov Yu. A., “Complexity of realization of a linear Boolean function in Sheffer’fs basis”,Moscow Univ.Math. Bull., 68:2, 114–117.10.3103/S0027132213020083Suche in Google Scholar
[11] Kombarov Yu. A., “On minimal circuts in Sheffer basis for linear Boolean functions”, J. Appl. Industr. Math., 20:4 (2013), 65–87 (in Russian).Suche in Google Scholar
[12] Redkin N. P., “On the complete fault detection tests for functional element circuits”, Matematicheskie voprosy kibernetiki, 1989,№2, 198–222 (in Russian).Suche in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Completeness criterion for the enumeration closure operator in three-valued logic
- Classification of distance-transitive orbital graphs of overgroups of the Jevons group
- Necessary conditions of applicability of Gaussian elimination to systems of equations over quasigroups
- The generalized complexity of linear Boolean functions
- On some implicitly precomplete classes of monotone functions in Pk
- Trees without twin-leaves with smallest number of maximal independent sets
- Limit theorems for some classes of power series type distributions
Artikel in diesem Heft
- Frontmatter
- Completeness criterion for the enumeration closure operator in three-valued logic
- Classification of distance-transitive orbital graphs of overgroups of the Jevons group
- Necessary conditions of applicability of Gaussian elimination to systems of equations over quasigroups
- The generalized complexity of linear Boolean functions
- On some implicitly precomplete classes of monotone functions in Pk
- Trees without twin-leaves with smallest number of maximal independent sets
- Limit theorems for some classes of power series type distributions
