Question

Design (specify ? for) a two-input perceptron (with an additional bias or offset term) that computes the following boolean functions. Assume T-1 and F. If a valid perceptron exists, show that it is not unique by designing another valid perceptron (with a different hyperplane, not simply through normalization). If no perceptron exists, state why a) AND (b) XOR

Answer 1

(a)

θ= (-1,1,1).

θ= (-1,0.5,0.6).

(b)

No perception exists

Step-by-step explanation:

For x= (1, x1, x2), a valid perception defined by

θ= (θ0, θ1, θ2) requires that y= sign(θ^T,x).

The possible values for AND and XOR operation are

[AND]

x1 x2 y

-1 -1 -1

-1 +1 -1

+1 -1 -1

+1 +1 +1

where y = x1 AND x2

[XOR]

x1 x2 y

-1 -1 -1

-1 +1 +1

+1 -1 +1

+1 +1 -1

Where y = x1 XOR x2

(a)

One possible perception is

θ= (-1,1,1).

Another possible perception is

θ= (-1,0.5,0.6).

There are other possible perceptions too

(b)

No perception exists because the data is not linearly separable.