Reliable networks from unreliable gates with almost minimal complexity

Abstract

We consider (combinatorial) networks constructed by using unreliable gates with a given error probability. We show that for almost all Boolean functions f there are networks realizing f, having almost the same error probability as the gates and having almost the same complexity as the minimal (unreliable) networks realizing f in case no gate has failed (having a very great error probability). This may be contrasted with results of 1.) von Neumann (1952), 2.) Dobrushin/Ortyukov (1977), 3.) Pippenger (1985) to the effect that the number of gates needed 1.) for minimal (reliable) networks is larger by at most a logarithmic factor than the number needed for unreliable networks [5], 2.) for some Boolean functions is larger by at least a logarithmic factor, 3.) for almost all Boolean functions is a (very great) multiple of the number of gates for unreliable realizations.