Reliable networks from unreliable gates with almost minimal complexity
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 , 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.
Unable to display preview. Download preview PDF.
- Dobrushin, R.L. and S.I. Ortyukov: Upper Bound for Redundancy of Self-Correcting Arrangements of Unreliable Functional Elements. Prob. of Info Transm. 13 (1977), 203–218.Google Scholar
- Dobrushin, R.L. and S.I. Ortyukov: On the Lower Bound for Redundancy of Self-Correcting Networks of Unreliable Functional Elements. Prob. Peredači Informacii 13 (1977) 1, 82–89. (Russian)Google Scholar
- Kirienko, G.I.: Synthesis of Combinatorial Networks which are Self-Correcting Referring to a Growing Number of Errors. Diskrenij analiz, Sb. Trudov IM SO AN SSSR 16 (1970) 38–43. (Russian)Google Scholar
- Lupanov, O.B.: On a Method of Synthesis of Networks. Izv. Vysš. Učebn. Zavad. Radiofizika 1 (1958) 1, 120–140. (Russian)Google Scholar
- Neumann von, J.: Probabilistic Logic of Reliable Organism from Unreliable Components. In: C.E. Shannon and J. McCarhy (Eds.), Automata studies, Princeton University Press (1956) 43–98.Google Scholar
- Ortyukov, S.I.: On the Synthesis of Asymptotically Nonredundant Self-Correcting Networks of Unreliable Functional Elements. Prob. Peredači Informacii 13 (1977) 4, 3–8. (Russian)Google Scholar
- Pippenger, N.: On Networks of Noisy Gates 26. Symposium on Foundation on Computer science, 21.–23. 10. 1985, Portland, 30–38.Google Scholar
- Savage, J.E.: The Complexity of Computing. Wiley-Interscience, New York, 1976.Google Scholar
- Uhlig, D.: On Reliable Networks from Unreliable Gates. Prepared for the Proceedings of the "International Workshop on Parallel Algorithms and Architectures" 25.–30. 5. 1987, Suhl.Google Scholar
- Uhlig, D.: Combinatorial Networks which are Self-Correcting and have almost the Smallest Complexity. Wiss. Beitraege der FSU Jena, Kompliziertheit von Lern-und Erkennungsprozessen (1975). 225–228. (German)Google Scholar
- Uhlig, D.: On the necessary proportion of reliable elements, Vortraege zur Automatentheorie, Weiterbildungszentrum fuer MKR der TU Dresden 6 (1974) 72–76. (German)Google Scholar