Computing Boolean Functions from Multiple Faulty Copies of Input Bits

Abstract

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 showt hat if f : 0, 1ⁿ → 0, 1 and ∈ 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 l _triv(f) the analogous number for the trivial construction, then l ^triv(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 D ^rand _stat,(f) = min∑ⁿ _i=1 l _i, where l _i is the number of 0.1-biased copies of x _i, 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 D ^rand _stat,(f) through a combinatorial lemma, that can be interesting on its own right.