C++ Neural Networks and Fuzzy Logic: Preface

InputActivationOutputNeeded Condition

0, 0000 < ¸

1, 0w

1

1w

1

 > ¸

0, 1w

2

1w

2

 > ¸

1, 1w

1

 + w

2

0w

1

 + w

2

< ¸

From the first three conditions, you can deduce that the sum of the two weights has to be greater than ¸, which

has to be positive itself. Line 4 is inconsistent with lines 1, 2, and 3, since line 4 requires the sum of the two

weights to be less than ¸. This affirms the contention that it is not possible to compute the XOR function with

C++ Neural Networks and Fuzzy Logic:Preface

XOR Function and the Perceptron

81

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a simple perceptron.

Geometrically, the reason for this failure is that the inputs (0, 1) and (1, 0) with which you want output 1, are

situated diagonally opposite each other, when plotted as points in the plane, as shown below in a diagram of

the output (1=T, 0=F):

 F     T

 T     F

You can’t separate the T’s and the F’s with a straight line. This means that you cannot draw a line in the plane

in such a way that neither (1, 1) −>F nor (0, 0)−>F is on the same side of the line as (0, 1) −>T and (1, 0)−> T.