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

Abstract

We study the following computational problem: for which values of $k$, the majority of $n$ bits ${\text{MAJ}}_n$ 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 ${\text{MAJ}}_k\circ {\text{MAJ}}_k$. We observe that the minimum value of $k$ for which there exists a ${\text{MAJ}}_k\circ {\text{MAJ}}_k$ circuit that has high correlation with the majority of $n$ bits is equal to $\Theta (n^{1/2})$. We then show that for a randomized ${\text{MAJ}}_k\circ {\text{MAJ}}_k$ 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 ${\text{MAJ}}_k\circ {\text{MAJ}}_k$ circuit computes the majority of $n$ bits correctly on all inputs, then $k\ge n^{13/19+o(1)}$. 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 ${\text{MAJ}}_n$ correctly on all inputs.