Computing Boolean Functions from Multiple Faulty Copies of Input Bits

• Authors: Mario Szegedy;Xiaomin Chen
• Venue: LATIN '02 Proceedings of the 5th Latin American Symposium on Theoretical Informatics
• Year: 2002

Suppose, we want to compute a Boolean function f, but instead of receiving the input, we only get l 驴-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 showt hat if f : {0, 1}n 驴 {0, 1} and 驴 are known, the best construction function, F, is often not the trivial. In particular, in many cases the best F cannot be written as a composition of some functions with f, 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) = 驴(l(f)). Moreover, both quantities are in 驴(log S(f)), where S(f) is the sensitivity of f.A quantity related to l(f) is Dstat,驴rand(f) = min驴i=1n li, where li is the number of 0.1-biased copies of xi, such that the above number of readings is already sufficient to recover f with high certainty. This quantity was first introduced by Reischuk et al. [14] in order to provide lower bounds for the noisy circuit size of f. In this article we give a complete characterization of Dstat,驴rand(f) through a combinatorial lemma, that can be interesting on its own right.