Unreliability of circuits in case of constant failures on inputs and outputs of gates

M. A. Alekhina

Prikladnaya Diskretnaya Matematika. Supplement, no. 8 (2015), pp. 100-102

Résumé

We consider the implementation of Boolean functions by circuits of unreliable functional elements in the basis containing only Sheffer stroke. It is assumed that each of the circuit elements is exposed to type 0 or type 1 failures in its inputs and outputs with probabilities $\gamma_0$ or $\gamma_1$ and $\varepsilon_0$ or $\varepsilon_1$ respectively. It is shown that any Boolean function can be so implemented by a such circuit that the asymptotic estimate of its unreliability is no more than $2\varepsilon_0+2\gamma_0+\varepsilon_1+2\gamma_1^2$ for $\gamma_0,\gamma_1,\varepsilon_0,\varepsilon_1\to0$. This estimation is achieved for functions $f\not\in\bigcup_{n=1}^\infty K(n)$ where $K(n)$ is the set of all Boolean functions $\bar x_i\vee h$ and $x_i\wedge\bar h$ for $i\in\{1,\dots,n\}$ and $h$ – an arbitrary Boolean function of variables $x_1,\dots,x_n$.

Détail

Keywords: unreliable functional gates, unreliability of circuits, constant failures.