Geodesic
About realizations of Boolean functions by asymptotically optimal reliable circuits
Diskretnyj analiz i issledovanie operacij, Tome 15 (2008) no. 6, pp. 63-89 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The following two problems are solved in this work: 1. It is shown that if each of irreducible complete base of functions of one or two variables is supplemented by one nonconstant Boolean function $\varphi(x_1,x_2)$ noncongruent to the basis functions, in most bases the estimation of reliability of functional elements circuits, subject to inverse malfunctions on the input gates of elements, reduces for almost all functions. 2. It is shown that if each of irreducible complete base of functions of one or two variables is supplemented by $k$ $(k\ge3)$ nonconstant Boolean functions noncongruent to the basis functions, then in all obtained bases the estimation of reliability of functional elements circuits, subject to inverse malfunctions on the input gates of elements, is asymptotically (as $\varepsilon\to0$) $2\varepsilon$ (i.e., trivial) for all functions $f(x_1,x_2,\dots,x_n)$ except constants 0 and 1 and functions $x_i$ and $\overline x_i$, where $\varepsilon$ is the failure probability at each input of functional element, $i=\overline{1,n}$. It is also shown that the circuits constructed while solving the last two problems are asymptotically optimal reliable circuits as $\varepsilon\to0$. Illustr. 3, tabl. 6, bibl. 5.
Keywords: boolean functions, asymptotically optimal reliable circuits. 
@article{DA_2008_15_6_a6,
     author = {V. V. Chugunova},
     title = {About realizations of {Boolean} functions by asymptotically optimal reliable circuits},
     journal = {Diskretnyj analiz i issledovanie operacij},
     pages = {63--89},
     year = {2008},
     volume = {15},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DA_2008_15_6_a6/}
}
TY  - JOUR
AU  - V. V. Chugunova
TI  - About realizations of Boolean functions by asymptotically optimal reliable circuits
JO  - Diskretnyj analiz i issledovanie operacij
PY  - 2008
SP  - 63
EP  - 89
VL  - 15
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/DA_2008_15_6_a6/
LA  - ru
ID  - DA_2008_15_6_a6
ER  - 
%0 Journal Article
%A V. V. Chugunova
%T About realizations of Boolean functions by asymptotically optimal reliable circuits
%J Diskretnyj analiz i issledovanie operacij
%D 2008
%P 63-89
%V 15
%N 6
%U http://geodesic.mathdoc.fr/item/DA_2008_15_6_a6/
%G ru
%F DA_2008_15_6_a6
V. V. Chugunova. About realizations of Boolean functions by asymptotically optimal reliable circuits. Diskretnyj analiz i issledovanie operacij, Tome 15 (2008) no. 6, pp. 63-89. http://geodesic.mathdoc.fr/item/DA_2008_15_6_a6/

[1] Alekhina M. A., “O nadezhnosti dvoistvennykh skhem”, Materialy XI shkoly-seminara “Sintez i slozhnost upravlyayuschikh sistem” (N. Novgorod, 20–25 noyabrya 2000 g.), Izd-vo mekh.-mat. f-ta MGU, M., 2000, 6–8

[2] Alekhina M. A., “Nizhnie otsenki nenadezhnosti skhem v nekotorykh bazisakh pri odnotipnykh konstantnykh neispravnostyakh na vkhodakh elementov”, Diskret. analiz i issled. operatsii. Ser. 1, 9:3 (2002), 3–28 | MR | Zbl

[3] Alekhina M. A., Chugunova V. V., “Ob asimptoticheski nailuchshikh po nadezhnosti skhemakh v bazise $\{\,\lor,{^-}\}$ pri inversnykh neispravnostyakh na vkhodakh elementov”, Diskret. analiz i issled. operatsii. Ser. 1, 13:4 (2006), 3–17 | MR

[4] Chugunova V. V., Sintez asimptoticheski optimalnykh po nadezhnosti skhem pri inversnykh neispravnostyakh na vkhodakh elementov, Dis. $\dots$ kand. fiz.-mat. nauk, Penzenskii gos. un-t, Penza, 2007, 110 pp.

[5] Chugunova V. V., “O nadezhnosti skhem v nekotorykh privodimykh polnykh bazisakh”, Izv. vuzov. Povolzhskii region. Fiz.-mat. nauki, 2007, no. 2, 25–37; http:// www.pnzgu.ru/dep/k_dm/files/chugunova.pdf