Fuzzy Logic

Aisha**

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إنضم
11 نوفمبر 2010
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Fuzzy sets and fuzzy logic
We showed in the last chapter that the learning problem is​
NP-complete
for a broad class of neural networks. Learning algorithms may require an
exponential number of iterations with respect to the number of weights until
a solution to a learning task is found. A second important point is that in
backpropagation networks, the individual units perform computations more
general than simple threshold logic. Since the output of the units is not limited
to the values 0 and 1, giving an interpretation of the computation performed
by the network is not so easy. The network acts like a black box by computing a
statistically sound approximation to a function known only from a training set.
In many applications an interpretation of the output is necessary or desirable.
In all such cases the methods of
fuzzy logic can be used.

11.1.1 Imprecise data and imprecise rules​
Fuzzy logic can be conceptualized as a generalization of classical logic. Modern
fuzzy logic was developed by Lotfi Zadeh in the mid-1960s to model those
problems in which imprecise data must be used or in which the rules of inference
are formulated in a very general way making use of diffuse categories
[170]. In fuzzy logic, which is also sometimes called diffuse logic, there are not
just two alternatives but a whole continuum of truth values for logical propositions.
A proposition​
A can have the truth value 0.4 and its complement Ac

the truth value 0​
.5. According to the type of negation operator that is used,
the two truth values must not be necessarily add up to 1.
Fuzzy logic has a weak connection to probability theory. Probabilistic
methods that deal with imprecise knowledge are formulated in the Bayesian
framework [327], but fuzzy logic does not need to be justified using a probabilistic
approach. The common route is to generalize the findings of multivalued
logic in such a way as to preserve part of the algebraic structure [62]. In

R. Rojas: Neural Networks, Springer-Verlag, Berlin, 1996​
R. Rojas: Neural Networks, Springer-Verlag, Berlin, 1996​
290 11 Fuzzy Logic​
this chapter we will show that there is a strong link between set theory, logic,
and geometry. A fuzzy set theory corresponds to fuzzy logic and the semantic
of fuzzy operators can be understood using a geometric model. The geometric
visualization of fuzzy logic will give us a hint as to the possible connection
with neural networks.
Fuzzy logic can be used as an interpretation model for the properties of
neural networks, as well as for giving a more precise description of their performance.
We will show that fuzzy operators can be conceived as generalized
output functions of computing units. Fuzzy logic can also be used to specify
networks directly without having to apply a learning algorithm. An expert
in a certain field can sometimes produce a simple set of control rules for a
dynamical system with less effort than the work involved in training a neural
network. A classical example proposed by Zadeh to the neural network
community is developing a system to park a car. It is straightforward to formulate
a set of fuzzy rules for this task, but it is not immediately obvious
how to build a network to do the same nor how to train it. Fuzzy logic is now
being used in many products of industrial and consumer electronics for which
a​
good control system is sufficient and where the question of optimal control
does not necessarily arise.

11.1.2 The fuzzy set concept​
The difference between crisp (i.e., classical) and fuzzy sets is established by
introducing a​
membership function. Consider a finite set X = {x1, x2, . . . , xn}

which will be considered the universal set in what follows. The subset​
A of

X​
consisting of the single element x1 can be described by the n-dimensional
membership vector
Z(A) = (1, 0, 0, . . . , 0), where the convention has been
adopted that a 1 at the
i-th position indicates that xi belongs to A. The set

B​
composed of the elements x1 and xn is described by the vector Z(B) =
(1
, 0, 0, ..., 1). Any other crisp subset of X can be represented in the same way
by an
n-dimensional binary vector. But what happens if we lift the restriction
to binary vectors? In that case we can define the
fuzzy set C with the following
vector description:

Z​
(C) = (0.5, 0, 0, ..., 0)
In classical set theory such a set cannot be defined. An element belongs to
a subset or it does not. In the theory of fuzzy sets we make a generalization
and allow descriptions of this type. In our example the element
x1 belongs to
the set
C only to some extent. The degree of membership is expressed by a
real number in the interval [0
, 1], in this case 0.5. This interpretation of the
degree of membership is similar to the meaning we assign to statements such
as “person
x1 is an adult”. Obviously, it is not possible to define a definite
age which represents the absolute threshold to enter into adulthood. The act
of becoming mature can be interpreted as a continuous process in which the
membership of a person to the set of adults goes slowly from 0 to 1.

R. Rojas: Neural Networks, Springer-Verlag, Berlin, 1996​
R. Rojas: Neural Networks, Springer-Verlag, Berlin, 1996​
11.1 Fuzzy sets and fuzzy logic 291​
There are many other examples of such diffuse statements. The concepts
“old” and “young” or the adjectives “fast” and “slow” are imprecise but easy
to interpret in a given context. In some applications, such as expert systems,
for example, it is necessary to introduce formal methods capable of dealing
with such expressions so that a computer using rigid Boolean logic can still
process them. This is what the theory of fuzzy sets and fuzzy logic tries to​
accomplish.
 

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سبحان الله و بحمده سبحان الله العظيم
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جزاكم الله خيرا
 
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