Computing Boolean functions from multiple faulty copies of input bits

• Authors: Mario Szegedy;Xiaomin Chen
• Venue: Theoretical Computer Science - Latin American theorotical informatics
• Year: 2004

Suppose we want to compute a Boolean function f, but instead of receiving the input, we only get l ε-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 ε 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)=Θ(l(f)). Moreover, both quantities are in Θ(log S(f)), where S(f) is the sensitivity of f.A quantity related to l(f) is Dstat.εrand(f) = min Σi = 1n li where li is the number of 1/10-biased copies of xi such that the above number of readings is sufficient to recover f with high probability. This quantity was first 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 Dstat.εrand(f) through a combinatorial lemma that can be interesting on its own right.