Amplification of probabilistic boolean formulas

The amplification of probabilistic Boolean formulas refers to combining independent copies of such formulas to reduce the error probability. Les Valiant used the amplification method to produce monotone Boolean formulas of size O(n5.3) for the majority function of n variables. In this paper we show that the amount of amplification that Valiant obtained is optimal. In addition, using the amplification method we give an O(k4.3 n log n) upper bound for the size of monotone formulas computing the kth threshold function of n variables.