On Bases with Unreliability Coefficient~$2$

M. A. Alekhina ; A. V. Vasin

Matematičeskie zametki, Tome 95 (2014) no. 2, pp. 170-201

Résumé

Consider the realization of Boolean functions by networks from unreliable functional components in a complete basis $B\subset B_3$ ($B_3$ is the set of all Boolean functions depending on the variables $x_1$, $x_2$, $x_3$). It is assumed that all the components of the network are subject to inverse faults at the outputs independently of each other with probability $\varepsilon\in(0,1/2)$. In $B_3$, we obtain all complete bases in which the following two conditions simultaneously hold: 1) any function can be realized by a network with unreliability asymptotically not greater than $2\varepsilon$ ($\varepsilon\to 0$); 2) there exist functions (denote their set by $K$) that cannot be realized by networks with unreliability asymptotically less than $2\varepsilon$, $\varepsilon\to 0$. Such bases will be called bases with unreliability coefficient $2$. It is also proved that the set $K$ contains almost all functions.

Détail

Keywords: synthesis of reliable networks from unreliable components, Boolean function, complete basis, unreliability coefficient, error probability of a network, reliability-based optimal network, inverse faults of components, von Neumann iterative method, upper (lower) bound for the unreliability of a network.