C++ Neural Networks and Fuzzy Logic: Preface

Recall for the Example

If we input the fit vector (0.3, 0.7, 0.4, 0.2), the output (b

1

, b

2

, b

3

) is determined as follows, using b

j

 = max(

min(a

1

, w

1j

), …, min(a

m

, w

mj

), where m is the dimension of the ‘a’ fit vector, and w

ij

 is the ith row, jth column

element of the matrix W.

     b

1

 = max(min(0.3, 0.3), min(0.7, 0.4), min(0.4, 0.4),

          min(0.2, 0.2)) =  max(0.3, 0.4, 0.4, 0.2) = 0.4

     b

2

 = max(min(0.3, 0.3), min(0.7, 0.3), min(0.4, 0.3),

          min(0.2, 0.2)) = max( 0.3, 0.3, 0.3, 0.2 ) = 0.3

     b3 = max(min(0.3, 0.3), min(0.7, 0.7), min(0.4, 0.4),

          min(0.2, 0.2)) = max (0.3, 0.7, 0.4, 0.2) = 0.7

The output vector (0.4, 0.3, 0.7) is not the same as the second fit vector used, namely (0.4, 0.3, 0.9), but it is a

subset of it, so the recall is not perfect. If you input the vector (0.4, 0.3, 0.7) in the opposite direction, using

the transpose of the matrix W, the output is (0.3, 0.7, 0.4, 0.2), showing resonance. If on the other hand you

input (0.4, 0.3, 0.9) at that end, the output vector is (0.3, 0.7, 0.4, 0.2), which in turn causes in the other

direction an output of (0.4, 0.3, 0.7) at which time there is resonance. Can we foresee these results? The

following section explains this further.

Recall

Let us use the operator o to denote max–min composition. Perfect recall occurs when the weight matrix is

obtained using the max–min composition of fit vectors U and V as follows: