C++ Neural Networks and Fuzzy Logic: Preface

Example of the Cube Revisited

Let us return to the example of the cube with vertices at the origin O, and the points labeled A, B, C, D, E, F,

and G. Suppose the set of vertices O, A, F, and G give a value of 1 for the function to be evaluated, and the

other vertices give a –1. The two sets are not linearly separable as mentioned before. A simple Perceptron

cannot evaluate this function.

Can the addition of another layer of neurons help? The answer is yes. What would be the role of this

additional layer? The answer is that it will do the final processing for the problem after the previous layer has

done some preprocessing. This can do two separations in the sense that the set of eight vertices can be

separated—or partitioned—into three separable subsets. If this partitioning can also help collect within each

subset, like vertices, meaning those that map onto the same value for the function, the network will succeed in

its task of evaluating the function when the aggregation and thresholding is done at the output neuron.

Strategy

So the strategy is first to consider the set of vertices that give a value of +1 for the function and determine the

minimum number of subsets that can be identified to be each separable from the rest of the vertices. It is

evident that since the vertices O and A lie on one edge of the cube, they can form one subset that is separable.

The other two vertices, viz., F and one for G, which correspond to the value +1 for the function, can form a

second subset that is separable, too. We need not bother with the last four vertices from the point of view of

further partitioning that subset. It is clear that one new layer of three neurons, one of which fires for the inputs

corresponding to the vertices O and A, one for F, and G, and the third for the rest, will then facilitate the

correct evaluation of the function at the output neuron.

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C++ Neural Networks and Fuzzy Logic:Preface

Example of the Cube Revisited

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