C++ Neural Networks and Fuzzy Logic: Preface

C++ Neural Networks and Fuzzy Logic

by Valluru B. Rao

MTBooks, IDG Books Worldwide, Inc.

ISBN: 1558515526   Pub Date: 06/01/95

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A Second Look at the XOR Function: Multilayer Perceptron

By introducing a set of cascaded Perceptrons, you have a Perceptron network, with an input layer, middle or

hidden layer, and an output layer. You will see that the multilayer Perceptron can evaluate the XOR function

as well as other logic functions (AND, OR, MAJORITY, etc.). The absence of the separability that we talked

about earlier is overcome by having a second stage, so to speak, of connection weights.

You need two neurons in the input layer and one in the output layer. Let us put a hidden layer with two

neurons. Let w

11

, w

12

, w

21

, and w

22

, be the weights on connections from the input neurons to the hidden layer

neurons. Let v

1

, v

2

 , be the weights on the connections from the hidden layer neurons to the outout neuron.

We will select the w’s (weights) and the threshold values ¸

1

 , and ¸

2

 at the hidden layer neurons, so that the

input (0, 0) generates the output vector (0, 0), and the input vector (1, 1) generates (1, 1), while the inputs (1,

0) and (0, 1) generate (0, 1) as the hidden layer output. The inputs to the output layer neurons would be from

the set {(0, 0), (1, 1), (0, 1)}. These three vectors are separable, with (0, 0), and (1, 1) on one side of the

separating line, while (0, 1) is on the other side.

We will select the s (weights) and Ä, the threshold value at the output neuron, so as to make the inputs (0, 0)

and (1, 1) cause an output of 0 for the network, and an output of 1 is caused by the input (0, 1). The network

layout within the labels of weights and threshold values inside the nodes representing hidden layer and output

neurons is shown in Figure 5.1a. Table 5.2 gives the results of operation of this network.