An Average-Case Depth Hierarchy Theorem for Boolean Circuits

@article{Rossman2015AnAD,
  title={An Average-Case Depth Hierarchy Theorem for Boolean Circuits},
  author={Benjamin Rossman and Rocco A. Servedio and Li-Yang Tan},
  journal={2015 IEEE 56th Annual Symposium on Foundations of Computer Science},
  year={2015},
  pages={1030-1048}
}
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 ≥ 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) circuit that agrees with f on (1/2 + on(1)) fraction of all inputs must have size exp(nΩ(1/d)). This answers an open question posed by Hastad in his Ph.D. thesis [Has86b]. Our average-case depth… 

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