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Electronic Colloquium on Computational Complexity

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Reports tagged with Random projections:
TR15-065 | 18th April 2015
Benjamin Rossman, Rocco Servedio, Li-Yang Tan

An average-case depth hierarchy theorem for Boolean circuits

We prove an average-case depth hierarchy theorem for Boolean circuits over the standard basis of AND, OR, and NOT gates. Our hierarchy theorem says that for every $d \geq 2$, there is an explicit $n$-variable Boolean function $f$, computed by a linear-size depth-$d$ formula, which is such that any depth-$(d-1)$ ... more >>>

TR22-087 | 8th June 2022
Pooya Hatami, William Hoza, Avishay Tal, Roei Tell

Depth-$d$ Threshold Circuits vs. Depth-$(d + 1)$ AND-OR Trees

For $n \in \mathbb{N}$ and $d = o(\log \log n)$, we prove that there is a Boolean function $F$ on $n$ bits and a value $\gamma = 2^{-\Theta(d)}$ such that $F$ can be computed by a uniform depth-$(d + 1)$ $\text{AC}^0$ circuit with $O(n)$ wires, but $F$ cannot be computed ... more >>>

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