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

(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.