Computing Boolean Functions from Multiple Faulty Copies of Input Bits

Suppose we want to compute a Boolean function f, but instead of receiving the input, we only get l -faulty copies of each input bit. A typical solution in this case is to take the majority value of the faulty bits for each individual input bit and apply f on the majority values. We call this the trivial construction. We show that if f : {0; 1}n → {0; 1} and are known, the best function construction, F , is often not the trivial one. In particular, in many cases the best F cannot be written as a composition of f with some functions, and in addition it is better to use a randomized F than a deterministic one. We also prove that the trivial construction is optimal in some rough sense: if we denote by l(f) the number of 1 10 -biased copies we need from each input to reliably compute f using the best (randomized) recovery function F , and we denote by ltriv(f) the analogous number for the trivial construction, then ltriv(f)= (l(f)). Moreover, both quantities are in (log S(f)), where S(f) is the sensitivity of f. A quantity related to l(f) is D stat; (f) = min ∑n i=1 li, where li is the number of 1 10 -biased copies of xi such that the above number of readings is su7cient to recover f with high probability. This quantity was 8rst introduced by Reischuk and Schmeltz [14] in order to provide lower bounds for the noisy circuit size of f. In this article we give a complete characterization of D stat; (f) through a combinatorial lemma that can be interesting on its own right. c © 2004 Elsevier B.V. All rights reserved.