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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Chapter 9

FAM: Fuzzy Associative Memory

Introduction

In this chapter, you we will learn about fuzzy sets and their elements, both for input and output of an

associative neural network. Every element of a fuzzy set has a degree of membership in the set. Unless this

degree of membership is 1, an element does not belong to the set (in the sense of elements of an ordinary set

belonging to the set). In a neural network of a fuzzy system the inputs, the outputs, and the connection weights

all belong fuzzily to the spaces that define them. The weight matrix will be a fuzzy matrix, and the activations

of the neurons in such a network have to be determined by rules of fuzzy logic and fuzzy set operations.

An expert system uses what are called crisp rules and applies them sequentially. The advantage in casting the

same problem in a fuzzy system is that the rules you work with do not have to be crisp, and the processing is

done in parallel. What the fuzzy systems can determine is a fuzzy association. These associations can be

modified, and the underlying phenomena better understood, as experience is gained. That is one of the reasons

for their growing popularity in applications. When we try to relate two things through a process of trial and

error, we will be implicitly and intuitively establishing an association that is gradually modified and perhaps

bettered in some sense. Several fuzzy variables may be present in such an exercise. That we did not have full

knowledge at the beginning is not a hindrance; there is some difference in using probabilities and using fuzzy

logic as well. The degree of membership assigned for an element of a set does not have to be as firm as the

assignment of a probability.

The degree of membership is, like a probability, a real number between 0 and 1. The closer it is to 1, the less

ambiguous is the membership of the element in the set concerned. Suppose you have a set that may or may

not contain three elements, say, a, b, and c. Then the fuzzy set representation of it would be by the ordered

triple (m

a

, m

b

, m

c

), which is called the fit vector, and its components are called fit values. For example, the

triple (0.5, 0.5, 0.5) shows that each of a, b, and c, have a membership equal to only one−half. This triple

itself will describe the fuzzy set. It can also be thought of as a point in the three−dimensional space. None of

such points will be outside the unit cube. When the number of elements is higher, the corresponding points

will be on or inside the unit hypercube.

It is interesting to note that this fuzzy set, given by the triple (0.5, 0.5, 0.5), is its own complement, something

that does not happen with regular sets. The complement is the set that shows the degrees of nonmembership.

The height of a fuzzy set is the maximum of its fit values, and the fuzzy set is said to be normal, if its height is

1. The fuzzy set with fit vector (0.3, 0.7, 0.4) has height 0.7, and it is not a normal fuzzy set. However, by

introducing an additional dummy component with fit value 1, we can extend it into a normal fuzzy set. The

desirability of normalcy of a fuzzy set will become apparent when we talk about recall in fuzzy associative

memories.

C++ Neural Networks and Fuzzy Logic:Preface

Chapter 9 FAM: Fuzzy Associative Memory

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The subset relationship is also different for fuzzy sets from the way it is defined for regular sets. For example,

if you have a fuzzy set given by the triple (0.3, 0.7, 0.4), then any fuzzy set with a triple (a, b, c) such that a d

0.3, b d 0.7, and c d 0.4, is its fuzzy subset. For example, the fuzzy set given by the triple (0.1, 0.7, 0) is a

subset of the fuzzy set (0.3, 0.7, 0.4).