Shrinkage under Random Projections, and Cubic Formula Lower Bounds for $\mathbf{AC}^0$

Electron. Colloquium Comput. Complex., pp. 1802020.

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Abstract:

H\r{a}stad showed that any De Morgan formula (composed of AND, OR and NOT gates) shrinks by a factor of $O(p^{2})$ under a random restriction that leaves each variable alive independently with probability $p$ [SICOMP, 1998]. Using this result, he gave an $\widetilde{\Omega}(n^{3})$ formula size lower bound for the Andreev function, whic...More

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