# On reliable networks from unreliable gates

• Dietmar Uhlig
Communications
Part of the Lecture Notes in Computer Science book series (LNCS, volume 269)

## Abstract

The study of networks with small error probability was inaugurated by J.v. Neumann in [1] in 1952. Assume there is a complete set of (unreliable) gates with error probability ɛ smaller than 1/2. One of the central results in this field (argued heuristically by von Neumann and proved rigorously by R.L. Dobrushin and S.I. Ortyukov [5] in 1977) is the following: Let γ be any (extremely small) positive constant. Further let A be a network realizing Boolean function f provided no gate has failed. (The error probability of A may be very great.) Let C(A) be the complexity of A (i.e. the sum of "costs" of its elements). Then the function f can be realized by a (reliable) network Ã with error probability not greater than ɛ+γ and complexity O(C(A)logC(A)). N. Pippanger [2] showed in 1985 that for almost all Boolean functions f there are networks Ã with complexity O(C(A)) instead of O(C(A)logC(A)). The proof is very easy and ingenious. Note that O(C(A)) means the following: There is a constant c1 such that C(Ã)≤c1C(A). Constant c1 was not determined by Pippenger, but it seems to be very great. We show that there is a constant c2=c2(ɛ) such that C(Ã)≤c2C(A), where c2(ɛ) → 1 if ɛ → 0. Thus, for almost all Boolean functions networks constructed by our method have very small error probabilities (near that of the elements) and almost minimal complexity.

## Keywords

Input Vector Boolean Function Error Probability Small Positive Number Reliable Network
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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