Can Large Fanin Circuits Perform Reliable Computations in the Presence of Noise ?

For ordinary circuits with a xed upper bound on the maximal fanin of gates it has been shown that logarithmic redundancy is necessary and suucient 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 rst distinction the two most important models are circuits consisting of and-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 and/or{circuits and threshold circuits with unknown error probabilities. Gates with large fanin are of no use in this case. Circuits of arbitrary size, but xed 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.