Computing Majority by Constant Depth Majority Circuits with Low Fan-in Gates

We study the following computational problem: for which values of k , the majority of n bits MAJ n can be computed with a depth two formula whose each gate computes a majority function of at most k bits? The corresponding computational model is denoted by MAJ k ∘ MAJ k . We observe that the minimum value of k for which there exists a MAJ k ∘ MAJ k circuit that has high correlation with the majority of n bits is equal to Θ( n 1/2 ). We then show that for a randomized MAJ k ∘ MAJ k circuit computing the majority of n input bits with high probability for every input, the minimum value of k is equal to n 2/3 + o (1) . We show a worst case lower bound: if a MAJ k ∘ MAJ k circuit computes the majority of n bits correctly on all inputs, then k ≥ n 13/19 + o (1) .