Monotone Formulae For Majority
For n = 1 the median operator is just the unary identity operation x. For n = 3 the ternary median operator can be expressed using conjunction and disjunction as xy + yz + zx. Remarkably this expression denotes the same operation independently of whether the symbol + is interpreted as inclusive or or exclusive or.
For an arbitrary n there exists a monotone formula for majority of size O(n5.3) (Valiant 1984). This is proved using probabilistic method. Thus, this formula is non-constructive. However, one can obtain an explicit formula for majority of polynomial size using a sorting network of Ajtai, Komlós, and Szemerédi.
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