A generalization of Shannon function
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Nikolay P. Redkin
Abstract
When investigating the complexity of implementing Boolean functions, it is usually assumed that the basis inwhich the schemes are constructed and the measure of the complexity of the schemes are known. For them, the Shannon function is introduced, which associates with each Boolean function the least complexity of implementing this function in the considered basis. In this paper we propose a generalization of such a Shannon function in the form of an upper bound that is taken over all functionally complete bases. This generalization gives an idea of the complexity of implementing Boolean functions in the “worst” bases for them. The conceptual content of the proposed generalization is demonstrated by the example of a conjunction.
Originally published in Diskretnaya Matematika (2017) 29, №2, 70–83 (in Russian).
References
[1] Lupanov O. B., Asymptotic estimates for the complexity of control systems, Moscow State University Publishing House, Moscow, 1984 (in Russian), 136 pp.Suche in Google Scholar
[2] Yablonsky S.V., Introduction to Discrete Mathematics, Vyssh. shk., Moscow, 2003 (in Russian), 384 pp.Suche in Google Scholar
[3] Nechiporuk E.I., “On the complexity of schemes in some bases containing nontrivial elements with zero weights”, Problemy kibernetiki, 1962, 123–160 (in Russian).Suche in Google Scholar
[4] Lupanov O. B., “About one method of synthesis of schemes”, Izvestiya vuzov. Radiofizika, 1:1 (1958), 120–140 (in Russian).Suche in Google Scholar
[5] Redkin N.P., Discrete mathematics, Fizmatlit, Moscow, 2009 (in Russian).Suche in Google Scholar
[6] Soprunenko E.P., “On the minimal realization of some functions by scheme of functional elements”, Problemy kibernetiki, 1965, No 15, 117–134 (in Russian).Suche in Google Scholar
[7] Gorelik E.S., “On the complexity of the realization of elementary conjunctions and disjunctions in the basis {X/Y}”, Problemy kibernetiki, 1973, No 26, 27-36 (in Russian).Suche in Google Scholar
[8] Novikov S.V., Komissarov V.E., Suprun V.P. “The minimal realization of the function f=x1x2…xr by schemes from Sheffer type elements”, Vestn. Beloruss. un-ta, 1:2 (1975), 13-17 (in Russian).Suche in Google Scholar
[9] Kochergina G.A., “On the complexity of the realization of elementary conjunctions and disjunctions by schemes in some complete bases”, Matematicheskie voprosy kibernetiki, 2002,¹11, 219–246 (in Russian).Suche in Google Scholar
[10] Redkin N.P., “On the complete checking tests for schemes of functional elements”, Matematicheskie voprosy kibernetiki, 1989, No 2, 198–222 (in Russian).Suche in Google Scholar
[11] Redkin N.P., “The proof of minimality of some schemes of functional elements”, Problemy kibernetiki, 1970, No 23, 83–101 (in Russian).Suche in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- A subcritical decomposable branching process in a mixed environment
- Limit theorems for bounded branching processes
- On coincidences of tuples in a q-ary tree with random labels of vertices
- A generalization of Shannon function
- Reduced critical Bellman–Harris branching processes for small populations
- Estimates of the mean size of the subset image under composition of random mappings
- Necessary conditions for power commuting in finite-dimensional algebras over a field
Artikel in diesem Heft
- Frontmatter
- A subcritical decomposable branching process in a mixed environment
- Limit theorems for bounded branching processes
- On coincidences of tuples in a q-ary tree with random labels of vertices
- A generalization of Shannon function
- Reduced critical Bellman–Harris branching processes for small populations
- Estimates of the mean size of the subset image under composition of random mappings
- Necessary conditions for power commuting in finite-dimensional algebras over a field
