- Source: Lupanov representation
Lupanov's (k, s)-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.
Definition
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 2n−k 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
′
≤
s
{\displaystyle |A_{p}|=s'\leq s}
.
Let ƒi(x) = ƒ(x) iff x ∈ Ai.
Moreover, let
B
i
,
w
{\displaystyle B_{i,w}}
be the set of the columns whose intersection with
A
i
{\displaystyle A_{i}}
is
w
{\displaystyle w}
.
External links
Course material describing the Lupanov representation
An additional example from the same course material