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On reliability of circuits over the basis $$\{x \vee y \vee z, x \, \&\, y\, \& \, z, \bar x\}$$ under single-type constant faults at inputs of elements. (English. Russian original) Zbl 1121.94032
Discrete Math. Appl. 16, No. 2, 195-203 (2006); translation from Diskretn. Mat. 18, No. 1, 116-125 (2006).
Summary: We consider realisation of Boolean functions over the basis $$\{x \vee y \vee z, x\, \& \, y\, \& \, z, \bar x\}$$ by circuits of unreliable functional elements which are subject to single-type constant faults at inputs of the elements. Let $$\gamma$$ be the probability of a fault at an input of an element. By the unreliability of a circuit is meant the greatest probability of error at its output. In this paper, we find the asymptotically best realisation of an arbitrary Boolean function $$f(x _{1},\cdots, x_{n} )$$ such that the functions $$x_{i}$$ , $$i = 1, 2,\cdots ,n$$, are realised absolutely reliably, the constants 0 and 1 are realised as reliably as we wish, and the remaining functions are realised with unreliability asymptotically equal to $$\gamma ^{3}$$ as $$\gamma \rightarrow 0$$.
##### MSC:
 94C12 Fault detection; testing in circuits and networks
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##### References:
 [1] N. P. Redkin, Reliability and Diagnosis of Circuits. Moscow State Univ. Press, Moscow, 1992 (in Russian) [2] Math. Notes 20 pp 775– (1976) [3] J. von Neumann, Probabilistic logic and the synthesis of reliable organisms from unreliable parts. In: Automata Studies (C. E. Shannon, Ed.) Princeton Univ. Press, Princeton, 1956, pp. 43-98. [4] S. I. Ortyukov, On redundancy of realisation of Boolean functions by circuits of unreliable gates. In: Proc. Seminar Discrete Math. Appl. Moscow State Univ. Press, Moscow, 1989, pp. 166-168 (in Russian). [5] Lecture Notes Comput. Sci. 278 pp 462– (1987) [6] Discrete Math. Appl. 11 pp 493– (2001) [7] M. A. Alekhina, Lower bounds for unreliability of networks in some bases under single-type faults at the input of elements. Discrete Anal. Oper. Res., Ser. 1 (2002) 9, No. 3, 3-28 (in Russian). · Zbl 1029.68026
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