On reliable networks from unreliable gates
The study of networks with small error probability was inaugurated by J.v. Neumann in  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  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  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.
KeywordsInput Vector Boolean Function Error Probability Small Positive Number Reliable Network
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