Lupanov representation

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Lupanov's (ks)-representation, named after Oleg Lupanov, is a way of representing Boolean circuits so as to show that the reciprocal of the Shannon effect. Shannon had showed that almost all Boolean functions of n variables need a circuit of size at least 2nn−1. The reciprocal is that:

All Boolean functions of n variables can be computed with a circuit of at most 2nn−1 + o(2nn−1) gates.


The idea is to represent the values of a boolean function ƒ in a table of 2k rows, representing the possible values of the k first variables x1, ..., ,xk, and 2nk columns representing the values of the other variables.

Let A1, ..., Ap be a partition of the rows of this table such that for i < p, |Ai = s and |A_p|=s'\leq s. Let ƒi(x) = ƒ(x) iff x ∈ Ai.

Moreover, let B_{i,w} be the set of the columns whose intersection with A_i is w.

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