Noisy Computing of the Threshold Function

Let 𝖳𝖧_k denote the k-out-of-n threshold function: given n input Boolean variables, the output is 1 if and only if at least k of the inputs are 1. We consider the problem of computing the 𝖳𝖧_k function using noisy readings of the Boolean variables, where each reading is incorrect with some fixed and known probability p ∈ (0,1/2). As our main result, we show that, when k = o(n), it is both sufficient and necessary to use (1 ± o(1)) nlogk/δ/D_𝖪𝖫(p || 1-p) queries in expectation to compute the 𝖳𝖧_k function with a vanishing error probability δ = o(1), where D_𝖪𝖫(p || 1-p) denotes the Kullback-Leibler divergence between 𝖡𝖾𝗋𝗇(p) and 𝖡𝖾𝗋𝗇(1-p) distributions. In particular, this says that (1 ± o(1)) nlog1/δ/D_𝖪𝖫(p || 1-p) queries in expectation are both sufficient and necessary to compute the 𝖮𝖱 and 𝖠𝖭𝖣 functions of n Boolean variables. Compared to previous work, our result tightens the dependence on p in both the upper and lower bounds.