Abstract
We consider the realization of Boolean functions by asymptotically optimal reliable circuits with constant faults at the outputs of the gates in the basis {x|y} (where x|y п»- anticonjunction i.п». x|y = x&y). It is proved that almost all Boolean functions can be realized by asymptotically optimal reliable circuits that operate with unreliability asymptotically equal to 2e0 + e1 at e0, e1 → 0, where e0 - probability of faults of type 0 at the output of basis gate, e1 - probability of faults of type 1 at the output of basis gate.
Highlights
J.von Neumann [1] was the first who considered the problem of synthesis of reliable circuits from unreliable gates
He assumed that all gates of the circuit are exposed to inverse faults independently with probability ε (ε < 1/2), when functional gate with Boolean function e(x) in the faulty state realizes function e(x))
Using an iterative method J.von Neumann established, that any Boolean function can be realized by a circuit at ε < 1/6 for which the error’s probability at the output for any input set of variables is not greater than cε (c is the some constant, c depends from the basis)
Summary
J.von Neumann [1] was the first who considered the problem of synthesis of reliable circuits from unreliable gates. The problem of construction of asymptotically optimal reliable circuits in the basis {x|y} have been solved for various faults of gates (for example,at faults only of the type 0 at the outputs or only type of 1 at the outputs (briefly in [5], detail in [6]), at output inverse faults [7]) by the same method: at first it is constructed reliable circuits, their reliability is improved by some circuit (that generally depends on the considered basis), and it is proved an upper bound for the unreliability of circuits It is proved the lower bound for the unreliability of circuit realizing functions of a certain class K. The problem of constructing asymptotically optimal reliability circuits in which the gates are subjected to two types of faults with different probabilities, is considered the first time
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