About realizations of Boolean functions by asymptotically optimal reliable circuits

V. V. Chugunova

Diskretnyj analiz i issledovanie operacij, Tome 15 (2008) no. 6, pp. 63-89

Résumé

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.

Détail

Keywords: boolean functions, asymptotically optimal reliable circuits.