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 languageEnglish (US)
Pages (from-to)149-170
Number of pages22
JournalTheoretical Computer Science
Volume321
Issue number1
DOIs
StatePublished - Jun 16 2004
EventLatin American Theoretical Informatics - Cancun, Mexico
Duration: Apr 3 2002Apr 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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