## Abstract

Moore and Shannon have shown that relays with arbitrarily high reliability can be built from relays with arbitrarily poor reliability. Valiant used methods to construct monotone lead-once formulas of size O(n^{α+2}) (where α = log_{√5-1} 2 ≃ 3.27) that amplify (ψ - 1/n, ψ + 1/n) (where ψ + (√5 - 1)/2 ≃ 0.62) to (2^{-n}, 1 - 2^{-n}) and deduced as a cosequence 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 - 1/n, p + 1/n) to (1/4, 3/4) (where 0 < p < 1 is constant) has size Ω(n^{α}) and that any monotone, not necessarily read-once, contact network (and in particular any monotone formula) that amplifies (1/4, 3/4) to (2^{-n}, 1 - 2^{-n}) has size Ω(n^{2}). We 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 - 1/n, p + 1/n) to (2^{-n}, 1 - 2^{-n}) has size Ω(n^{α+2}). This result does not follow from Boppana's arguments, and its 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 nterworks of size O(n^{2.99}) that amplify (1/2 - 1/n, 1/2 + 1/n) to (1/4, 3/4). This shows that Boppana's lower bound for the first amplification once.

Original language | English |
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Pages (from-to) | 15-38 |

Number of pages | 24 |

Journal | SIAM Journal on Computing |

Volume | 26 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1997 |

## Keywords

- Amplification
- Boolean formula
- Circuit complexity