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 $\text{MAJ}_n$ 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 $Θ(n^{1/2})$. 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 $k\geq n^{13/19+o(1)}$. 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.

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