Abstract
A ∑II∑ formula has the form
, where eachL is either a variable or a negated variable. In this paper we study the computation of threshold functions by ∑II∑ formulas. By combining the proof of the Fredman-Komlós bound [5, 10] and a counting argument, we show that fork andn large andk≤n/2, every ∑II∑ formula computing the threshold functionT nk has size at least exp\((\Omega (\sqrt {k/\ln k} ))nlogn\). Fork andn large andk≤n 2/3, we show that there exist ∑II∑ formulas for computingT nk with size at most exp\((2(\sqrt {k/\ln k} ))nlogn\).
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