Amplification by read-once formulas

Moshe Dubiner*, Uri Zwick

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

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(nα+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(nα) 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(n2). 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(nα+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(n2.99) 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 languageEnglish
Pages (from-to)15-38
Number of pages24
JournalSIAM Journal on Computing
Volume26
Issue number1
DOIs
StatePublished - Feb 1997

Keywords

  • Amplification
  • Boolean formula
  • Circuit complexity

Fingerprint

Dive into the research topics of 'Amplification by read-once formulas'. Together they form a unique fingerprint.

Cite this