Computing Boolean functions from multiple faulty copies of input bits

Mario Szegedy, Xiaomin Chen

Abstract: Suppose we want to compute a Boolean function f, but instead of receiving the input, we only get lε<math><mtext>l</mtext><mspace xmlns="true" sp="0.33" width="2px" linebreak="nobreak" is="true"></mspace><mtext>ε</mtext></math>-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}<math><mtext>f</mtext><mspace xmlns="true" sp="0.16" width="2px" linebreak="nobreak" is="true"></mspace><mtext>:</mtext><mspace xmlns="true" sp="0.16" width="2px" linebreak="nobreak" is="true"></mspace><mtext>{0,1}</mtext><msup><mi></mi><mn>n</mn></msup><mtext>↦{0,1}</mtext></math> 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 110<math><mtext>1</mtext><mtext>10</mtext></math>-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))<math><mtext>Θ(</mtext><mtext>log</mtext><mspace xmlns="true" sp="0.16" width="2px" linebreak="nobreak" is="true"></mspace><mtext>S(f))</mtext></math>, where S(f) is the sensitivity of f.A quantity related to l(f) is Dstat,εrand(f)=min∑i=1nli<math><mtext>D</mtext><msub><mi></mi><mn><mtext>stat</mtext><mtext>,ε</mtext></mn></msub><msup><mi></mi><mn><mtext>rand</mtext></mn></msup><mtext>(f)=</mtext><mtext>min</mtext><mspace xmlns="true" sp="0.16" width="2px" linebreak="nobreak" is="true"></mspace><mtext>∑</mtext><msub><mi></mi><mn>i=1</mn></msub><msup><mi></mi><mn>n</mn></msup><mspace xmlns="true" sp="0.16" width="2px" linebreak="nobreak" is="true"></mspace><mtext>l</mtext><msub><mi></mi><mn>i</mn></msub></math>, where li is the number of 110<math><mtext>1</mtext><mtext>10</mtext></math>-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.