Analysis of Boolean Functions - Ch. 7 & 8 Solutions and Errata

7.2 Let M = {(x, y) ∈ {0, 1}2n : x = y}. Note that d(z,M) = 1 2n |{i ∈ [n] : zi 6= zn+i}|, so let’s have our local tester pick a uniformly random j ∼ [n] and accept iff zj = zn+j . The completeness of this test is obviously 1, and in general we see that z is accepted with probability equal to 1 − 1 n |{i ∈ [n] : zi 6= zn+i}| = 1 − 2d(z,M), so the rejection rate is actually 2. 7.3 We’ll still want our proof string Π to contain the partial sums Πj = ∑j+1 i=1 wj mod 2, but the last 2 partial sums are unnecessary, since we’re allowed to make 3 queries. Explicitly, let Π = (Π1, . . . ,Πn−3), and perform one of the following checks, uniformly at random: