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. 
@article{MZM_2011_89_3_a11,
     author = {V. V. Chugunova},
     title = {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}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {440--458},
     publisher = {mathdoc},
     volume = {89},
     number = {3},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2011_89_3_a11/}
}
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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/

[1] Dzh. Neiman, “Veroyatnostnaya logika i sintez nadezhnykh organizmov iz nenadezhnykh komponent”, Avtomaty, IL, M., 1956, 68–139

[2] M. A. Alekhina, Sintez asimptoticheski optimalnykh po nadezhnosti skhem, Monografiya, IITs PGU, Penza, 2006

[3] V. V. Chugunova, Sintez asimptoticheski optimalnykh po nadezhnosti skhem pri inversnykh neispravnostyakh na vkhodakh elementov, Dis. $\dots$ kand. fiz.-matem. nauk, Penza, 2007

[4] M. A. Alekhina, “O nadezhnosti skhem v bazise $\{\bigvee_{i=1}^kx_i,\bigwedge_{i=1}^kx_i,\overline x\}$ pri odnotipnykh konstantnykh neispravnostyakh na vkhodakh elementov”, Izv. vuzov. Povolzhsk. reg. Fiz.-matem. nauki, 2007, no. 1, 18–27