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Consider all functions $f : \{-1, 0, 1\}^3 \to \{-1, 1\}$. How many of these functions exist and how many can be realized by a perceptron.

What are the conditions that one has to check?

N.B. A perceptron is a function $f_{w,b} : {\bf R}^n \to {\bf R}$ such that $f_{w,b}(x) = {\rm sgn}(w\cdot x - b)$

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    What are perceptrons?2012-07-03
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    http://en.wikipedia.org/wiki/Perceptron2012-07-03
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    The number of possible functions is clearly $|\{-1,1\}|^{|\{-1,0,1\}^3|} = 2^{|\{-1,0,1\}|^3} = 2^{3^3} = 2^{27}$.2012-07-03
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    From the definition given, a perceptron corresponds to a plane in $\mathbb{R}^3$. The function then outputs, given an input point, whether the point is above or below the plane. So the number of perceptron functions is the same as twice (due to orientation) the number of ways the cube $\{-1,0,1\}^3$ can be cleft in twain.2012-07-03
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    @Makholm/Wong - How does one see how many of these possible functions can be realized by a perceptron?2012-07-03

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