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TR98-070 | 7th December 1998 00:00
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#### Can Large Fanin Circuits Perform Reliable Computations in the Presence of Noise?

**Abstract:**

For ordinary circuits with a fixed upper bound on the maximal fanin

of gates it has been shown that logarithmic redundancy is necessary and

sufficient to overcome random hardware faults.

Here, we consider the same question for unbounded fanin circuits that

in the noiseless case can compute Boolean functions in sublogarithmic depth.

In this case the details of the fault model become more important.

One may assume that only gates, resp. only wires may deliver wrong values,

or that both gates and wires may behave faulty due to random noise.

The fault tolerance depends on the types of gates that are used, and whether

the error probabilities are known exactly or only an upper bound for them.

Concerning the first distinction the two most important models are circuits

consisting of and/or-gates with arbitrarily many inputs,

and circuits built from the more general type of threshold gates.

We will show that reliable computation is basically impossible

for such circuits with unknown error probabilities.

Gates with large fanin are of no use in this case.

Circuits of arbitrary size, but fixed depth can compute

only a tiny subset of all Boolean functions reliably.

Only in case of threshold circuits and exactly known error probabilities

redundancy is able to compensate faults.

We describe a transformation from fault-free to fault-tolerant circuits

that is optimal with respect to depth keeping the circuit size polynomial.