A First Course in FUZZY and NEURAL CONTROL

5.2. IMPLEMENTING NEURAL NETWORKS 169

Figure 5.4. Two-layer neural network

mented by a perceptron, a neural network without hidden layers. Note that
with inputs(x 1 ,x 2 )pairs of real numbers, the domain is divided into two pieces
by the activation function

f(w 1 x 1 +w 2 x 2 −b)=

Ω

1 if w 1 x 1 +w 2 x 2 ≥b

0 if w 1 x 1 +w 2 x 2 <b

Namely, inputs above the linew 1 x 1 +w 2 x 2 −b=0produce an output of 1 ,and
inputs below that line produce an output of 0. In particular, this setup provides
a solution only if the underlying function that the neural network is trying to
implement islinearly separablein this sense.

Example 5.1We design a perceptron to implement the logical Boolean func-
tion OR, that is, the functiong:{ 0 , 1 }◊{ 0 , 1 }→{ 0 , 1 },defined by

g(x 1 ,x 2 )=

Ω

0 ifx 1 =x 2 =0

1 otherwise

In view of the functiong, we consider binary-input/binary-output neural net-
works, with two nodes in the input layer, and only one neuron in the output
layer.

y=f(w 1 x 1 +w 2 x 2 −b)
The problem is to choosew 1 ,w 2 ,andbso that

g(x 1 ,x 2 )=f(w 1 x 1 +w 2 x 2 −b)