### Computing Majority by Constant Depth Majority Circuits with Low Fan-in Gates [article]

Alexander S. Kulikov, Vladimir V. Podolskii
2016 arXiv   pre-print
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
more » ... 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). This lower bound exceeds the optimal value for randomized circuits and thus is unreachable for pure randomized techniques. For depth 3 circuits we show that a circuit with k= O(n^2/3) can compute MAJ_n correctly on all inputs.