Geodesic
Asymptotically Optimally Reliable Circuits in the Basis $\{x_1\mathbin{\} x_2\mathbin{\} x_3,x_1\vee x_2\vee x_3,\overline x_1\}$ for Inverse Faults at the Inputs of Elements
Matematičeskie zametki, Tome 89 (2011) no. 3, pp. 440-458

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We prove that, in the basis $\{x_1\mathbin{\&} x_2\mathbin{\&} x_3,x_1\vee x_2\vee x_3,\overline x_1\}$, for inverse faults at the inputs of functional elements, all Boolean functions $f(x_1,x_2,\dots,x_n)$ can be realized by asymptotically optimally reliable circuits operating with unreliability asymptotically (as $\varepsilon\to 0$) equal to: $\varepsilon^3$ for the constants $0$ and $1$, $\varepsilon$ for the functions $\overline x_1$, and $3\varepsilon$ for $f(x_1,x_2,\dots,x_n)\ne 0,1,\overline x_i,x_i$, where $\varepsilon$ is the error probability at each input of the functional element and $i=1,\dots,n$. The functions $x_i$, $i=1,\dots,n$, can be realized absolutely reliably. The complexity of asymptotically optimally reliable circuits is equal in order to the complexity of minimal circuits constructed only from reliable elements.
Keywords: optimally reliable circuit, unreliable functional element, inverse fault, Boolean function, AND gate, OR gate, voting function. 
V. V. Chugunova. Asymptotically Optimally Reliable Circuits in the Basis $\{x_1\mathbin{\&} x_2\mathbin{\&} x_3,x_1\vee x_2\vee x_3,\overline x_1\}$ for Inverse Faults at the Inputs of Elements. Matematičeskie zametki, Tome 89 (2011) no. 3, pp. 440-458. http://geodesic.mathdoc.fr/item/MZM_2011_89_3_a11/
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