Teaching Computer Science (CS 501)
Presentation Report
Introduction to Fuzzy Systems
TA: Mohammad Sadiq
Group members:
Amna Bilal (2003-02-0030)
Mehru Bint e Asad(2003-02-0104)
Abstract
Traditional logic has not led to a significant better understanding of
thought processes, concept formation, and pattern recognition,
because classic logic has not successfully captured uncertainty and
imprecision prevailing in the real world problems. This lecture
introduces fuzzy logic as a method to model vagueness, uncertainty
and imprecision, through structuring the input-output pairs as fuzzy ifthen rules that relate linguistic or fuzzy variables. This report will
concentrate on the appreciation of the need of fuzzy logic by
comparing with crisp/binary logic. Basic fuzzy set theory, fuzzy set
operations and an introduction to membership functions will form an
outline for this report.
Panoramic View
The real world is complex; complexity in the real world generally arises
from uncertainty in the form of ambiguity. Problems featuring
complexity and ambiguity have been addressed subconsciously by
humans since they could think. Why then are computers, which have
been designed by humans after all, not capable of addressing complex
and ambiguous issues? How can humans reason about real systems,
when the complete description of a real system often requires more
detailed data than a human could ever hope to recognize
simultaneously and assimilate with understanding? The answer is that
humans have the capacity to reason approximately, a capability that
computers currently do not have. In reasoning about a complex
system, humans reason approximately about its behavior, thereby
maintaining only a generic understanding about the problem.
Fuzzy logic and hence fuzzy systems is a domain of artificial
intelligence that captures real world problems more aptly than the
traditional crisp/binary/boolean logic does. Fuzzy logic is a superset of
conventional (Boolean) logic that has been extended to handle the
concept of partial truth -- truth values between "completely true" and
"completely false". This is contrary to the binary logic that computers
work on. Rather than regarding fuzzy theory as a single theory, we
should regard the process of “fuzzification'' as a methodology to
generalize ANY specific theory from a crisp (discrete) to a continuous
(fuzzy) form.
Just as there is a strong relationship between Boolean logic and the
concept of a subset, there is a similar strong relationship between
fuzzy logic and fuzzy subset theory.
In classical set theory, a subset U of a set S can be defined as a
mapping from the elements of S to the elements of the set {0, 1},
U: S --> {0, 1}
This mapping may be represented as a set of ordered pairs, with
exactly one ordered pair present for each element of S. The first
element of the ordered pair is an element of the set S, and the second
element is an element of the set {0, 1}. The value zero is used to
represent non-membership, and the value one is used to represent
membership. The truth or falsity of the statement
x is in U
is determined by finding the ordered pair whose first element is x. The
statement is true if the second element of the ordered pair is 1, and
the statement is false if it is 0.
Similarly, a fuzzy subset F of a set S can be defined as a set of ordered
pairs, each with the first element from S, and the second element from
the interval [0,1], with exactly one ordered pair present for each
element of S. This defines a mapping between elements of the set S
and values in the interval [0,1]. The value zero is used to represent
complete non-membership, the value one is used to represent
complete membership, and values in between are used to represent
intermediate DEGREES OF MEMBERSHIP. The set S is referred to as
the UNIVERSE OF DISCOURSE for the fuzzy subset F. Frequently, the
mapping is described as a function, the MEMBERSHIP FUNCTION of F.
The degree to which the statement
x is in F
is true is determined by finding the ordered pair whose first element is
x. The DEGREE OF TRUTH of the statement is the second element of
the ordered pair.
If we consider the age that categorizes when one is called “young”, we
have to classify it as a fuzzy set for it to make sense. Otherwise,
choosing an age arbitrarily and classifying it to be the benchmark until
which one is young and beyond which (be it a day) one is not young,
would not be a realistic judgment. Hence, we describe “young” as a
linguistic variable, which represents our cognitive category of
“youngness”. To each person in the universe of discourse, we have to
assign a degree of membership in the fuzzy subset YOUNG. The
easiest way to do this is with a membership function based on the
person's age. For example:
as opposed to the crisp case:-
In the first case, a person belongs to the category of young people
with a certain degree of membership, whereas in the crisp (second)
case a person either belongs to the category of young people or does
not.
In fuzzy logic, the membership function can take other shapes as well
depending upon the knowledge obtained about the set and its
members and depending upon the criterion being used.
The standard fuzzy logic operators are:truth (not x) = 1.0 - truth (x)
truth (x and y) = minimum (truth(x), truth(y))
truth (x or y) = maximum (truth(x), truth(y))
where; x and y are fuzzy variables.
For fuzzy logic the Law of Excluded Middles does not hold true as it
does for crisp logic. The union of a fuzzy set and its complement does
not give the entire universe of discourse. Also, the intersection of a
fuzzy set and its complement is not a null set as it is in the crisp case.
Critical Stages of Understanding
The critical stages of understanding are:1. To be able to create a feeling of appreciation in the audience of
the meaning of ‘fuzziness’ as opposed to ‘crisp’ by using simple
examples to elucidate the difference and hence the need of
having some other form of logic to represent degrees.
2. An important stage of understanding is the concept of
membership to a set and hence membership function and
shapes. The student should be excited enough to explore all
possible shapes of membership functions for a given problem.
3. The student should be able to clearly differentiate between crisp
and fuzzy logic and should be curious to delve into the
applications of fuzzy logic on his/her own. The student should
also be able to recognize and comprehend the absence of the
Law of Excluded Middles in the case of fuzzy logic.
Critical Operations
The following operations are important to the development of this
lecture.
1. Union
2. Intersection
3. Complement
All these three operations are vital in explaining the concept of the Law
of Excluded Middles.
Feedback Mechanisms

To keep the lecture alive and have a healthy interaction, getting
response from the audience is vital from time to time. It is also
important to monitor the pace of the lecture and ensure that the
audience is following whatever is being discussed. There are points
in the lecture when it is best to invite such interaction. One such
point should be when asking the audience for a benchmark that
they would like to suggest for the age of the young person and
from thereon argue with them why for example they categorize 27
as young and not 28, etc. This will lead to a situation where they
will be able to reflect upon the problem themselves and reach the
conclusion of the need of fuzziness on their own.

When the concept of having different shapes to membership
functions is introduced, the audience can be asked to give some
other shape than the one that is being illustrated at the time and
then justify their choice. This also helps in recapping and reinforcing
a point that is being made.

To keep curiosity alive and to have the audience restless and
thinking, we can end the lecture on a note when they are asked to
find out whether fuzzy logic is based more on data or knowledge.
Leaving this question unanswered and providing them with some
basic knowledge of fuzzy logic and some resources, which they can
access; this helps in initiating group discussion and individual
research and involvement of the student even after the class. This
also provides a link to take off from in the successive class or the
course.
Compared to what/Connected to what
Most of the lecture is based on a critical appraisal of the difference
between crisp and fuzzy logic, hence the comparison of fuzzy logic
with crisp logic and the emphasis on the need of fuzzification with
respect to real world problems.
Slides Used
The transparencies were used in a part of the lecture to have a small
recap and reinforcement in between the lecture. It helped to give a
visual reinforcement to the audience of the difference between the
crisp and logic case for the same example.
Feedback from Audience (Drawbacks and Plus Points)