# An Average-Case Depth Hierarchy Theorem for Boolean Circuits

```@article{Hstad2015AnAD,
title={An Average-Case Depth Hierarchy Theorem for Boolean Circuits},
author={Johan H{\aa}stad and Benjamin Rossman and Rocco A. Servedio and Li-Yang Tan},
journal={Electronic Colloquium on Computational Complexity (ECCC)},
year={2015},
volume={22},
pages={65}
}```
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 <i>d</i> ≥ 2, there is an explicit <i>n</i>-variable Boolean function <i>f</i>, computed by a linear-size depth-<i>d</i> formula, which is such that any depth-(<i>d</i>−1) circuit that agrees with <i>f</i> on (1/2 + <i>o</i><sub><i>n</i></sub>(1)) fraction of all inputs must have size exp(<i>n</i><sup>Ω (1/d)</sup>). This answers an… CONTINUE READING
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