# Computing Boolean functions from multiple faulty copies of input bits

Mario Szegedy, Xiaomin Chen

Research output: Contribution to journalConference articlepeer-review

2 Scopus citations

## 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 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 110-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(logS(f)), where S(f) is the sensitivity of f. A quantity related to l(f) is Dstat,rand(f)=min∑ i=1nli, where li is the number of 110-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,rand(f) through a combinatorial lemma that can be interesting on its own right.

Original language English (US) 149-170 22 Theoretical Computer Science 321 1 https://doi.org/10.1016/j.tcs.2003.07.001 Published - Jun 16 2004 Latin American Theoretical Informatics - Cancun, MexicoDuration: Apr 3 2002 → Apr 6 2002

## All Science Journal Classification (ASJC) codes

• Theoretical Computer Science
• Computer Science(all)

## Keywords

• Boolean functions
• Linear programming
• Randomized functions
• Sensitivity

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