Amplification and percolation [probabilistic Boolean functions]

The authors extend R.B. Boppana´s results (1989) in two ways. They first show that his two lower bounds hold for general read-once formulae, not necessarily monotone, that may even include exclusive-or gates. They 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 of at least Ω(n^α+2). This result does not follow from Boppana´s arguments and it shows that the amount of amplification achieved by L.G. Valiant (1984) is the maximal achievable using read-once formulae