Amplification by Read-Once Formulas

Abstract

Moore and Shannon have shown that relays with arbitrarily high reliability can be built from relays with arbitrarily poor reliability. Valiant used similar methods to construct monotone read-once formulas of size $O(n^{\alpha+2})$ (where $\alpha=\log_{\sqrt{5}-1}2\simeq 3.27$) that amplify $(\psi-\frac{1}{n},\psi+\frac{1}{n})$ (where $\psi=(\sqrt{5}-1)/2\simeq0.62$) to $(2^{-n},1-2^{-n})$ and deduced as a consequence the existence of monotone formulas of the same size that compute the majority of $n$ bits. Boppana has shown that any monotone read-once formula that amplifies $(p-\frac{1}{n},p+\frac{1}{n})$ to $(\frac{1}{4},\frac{3}{4})$ (where $0We extend Boppana's results in two ways. We first show that his two lower bounds hold for general read-once formulas, not necessarily monotone, that may even include exclusive-or gates. We are then able to join his two lower bounds together and show that any read-once, not necessarily monotone, formula that amplifies $(p-\frac{1}{n},p+\frac{1}{n})$ to $(2^{-n},1-2^{-n})$ has size $\Omega(n^{\alpha+2})$. This result does not follow from Boppana's arguments, and it shows that the amount of amplification achieved by Valiant is the maximal achievable using read-once formulas. In a companion paper we construct monotone read-once contact networks of size $O(n^{2.99})$ that amplify $(\frac{1}{2}-\frac{1}{n},\frac{1}{2}+\frac{1}{n})$ to $(\frac{1}{4},\frac{3}{4})$. This shows that Boppana's lower bound for the first amplification stage does not apply to contact networks, even if they are required to be both monotone and read-once.