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Fuzzy Logic

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Fuzzy Logic * * * * * * * * WHAT IS FUZZY LOGIC? Definition of fuzzy Fuzzy not clear, distinct, or precise; blurred Definition of fuzzy logic A form of ... – PowerPoint PPT presentation

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Title: Fuzzy Logic


1
Fuzzy Logic
2
WHAT IS FUZZY LOGIC?
  • Definition of fuzzy
  • Fuzzy not clear, distinct, or precise
    blurred
  • Definition of fuzzy logic
  • A form of knowledge representation suitable for
    notions that cannot be defined precisely, but
    which depend upon their contexts.

3
Fuzziness Vs. Vagueness
3
4
TRADITIONAL REPRESENTATION OF LOGIC
Slow
Fast
Speed 0
Speed 1
bool speed get the speed if ( speed 0)
// speed is slow else // speed is
fast
5
FUZZY LOGIC REPRESENTATION
Slowest
  • Every problem must represent in terms of fuzzy
    sets.

0.0 0.25
Slow
0.25 0.50
Fast
0.50 0.75
Fastest
0.75 1.00
6
FUZZY LOGIC REPRESENTATION CONT.
Slowest
Fastest
Slow
Fast
float speed get the speed if ((speed gt
0.0)(speed lt 0.25)) // speed is slowest
else if ((speed gt 0.25)(speed lt 0.5)) //
speed is slow else if ((speed gt 0.5)(speed lt
0.75)) // speed is fast else // speed gt
0.75 speed lt 1.0 // speed is fastest
7
ORIGINS OF FUZZY LOGIC
  • Lotfi Asker Zadeh ( 1965 )
  • First to publish ideas of fuzzy logic.
  • Professor Toshire Terano ( 1972 )
  • Organized the world's first working group on
    fuzzy systems.
  • F.L. Smidth Co. ( 1980 )
  • First to market fuzzy expert systems.

8
TEMPERATURE CONTROLLER
  • The problem
  • Change the speed of a heater fan, based on the
    room temperature and humidity.
  • A temperature control system has four settings
  • Cold, Cool, Warm, and Hot
  • Humidity can be defined by
  • Low, Medium, and High
  • Using this we can define
  • the fuzzy set.

9
Introduction
  • Fuzzy Logic is used to provide mathematical rules
    and functions which permitted natural language
    queries.
  • Fuzzy logic provides a means of calculating
    intermediate values between absolute true and
    absolute false with resulting values ranging
    between 0.0 and 1.0.
  • With fuzzy logic, it is possible to calculate the
    degree to which an item is a member.

10
  • For example, if a person is .83 of tallness, they
    are rather tall.
  • Fuzzy logic calculates the shades of gray between
    black/white and true/false.
  • Fuzzy logic is a super set of conventional (or
    Boolean) logic and contains similarities and
    differences with Boolean logic.
  • Fuzzy logic is similar to Boolean logic, in that
    Boolean logic results are returned by fuzzy logic
    operations when all fuzzy memberships are
    restricted to 0 and 1.
  • Fuzzy logic differs from Boolean logic in that it
    is permissive of natural language queries and is
    more like human thinking it is based on degrees
    of truth.

11
Fuzzy Logic
Boolean Logic
12
  • Fuzzy logic may appear similar to probability and
    statistics as well.
  • Although, fuzzy logic is different than
    probability even though the results appear
    similar.
  • The probability statement, " There is a 70
    chance that Bill is tall" supposes that Bill is
    either tall or he is not. There is a 70 chance
    that we know which set Bill belongs.
  • The fuzzy logic statement, " Bill's degree of
    membership in the set of tall people is .80 "
    supposes that Bill is rather tall.
  • The fuzzy logic answer determines not only the
    set which Bill belongs, but also to what degree
    he is a member.

13
  • Fuzzy logic deals with the degree of membership.
  • Fuzzy logic has been applied in many areas it is
    used in a variety of ways.
  • Household appliances such as dishwashers and
    washing machines use fuzzy logic to determine the
    optimal amount of soap and the correct water
    pressure for dishes and clothes.
  • Fuzzy logic is even used in self-focusing
    cameras.
  • Expert systems, such as decision-support and
    meteorological systems, use fuzzy logic.

14
History
  • Fuzzy Logic deals with those imprecise conditions
    about which a true/false value cannot be
    determined.
  • Much of this has to do with the vagueness and
    ambiguity that can be found in everyday life.
  • For example, the question Is it HOT outside?
  • These are often labelled as subjective responses,
    where no one answer is exact.
  • Subjective responses are relative to an
    individual's experience and knowledge.
  • Human beings are able to exert this higher level
    of abstraction during the thought process.

15
  • For this reason, Fuzzy Logic has been compared to
    the human decision making process.
  • Conventional Logic (and computing systems for
    that matter) are by nature related to the Boolean
    Conditions (true/false).
  • What Fuzzy Logic attempts to encompass is that
    area where a partial truth can be established.
  • In fuzzy set theory, although it is still
    possible to have an exact yes/no answer as to set
    membership, elements can now be partial members
    in a set.

16
Fuzzy Sets
  • Fuzzy logic is a superset of Boolean
    (conventional) logic that handles the concept of
    partial truth, which is truth values between
    "completely true" and "completely false".
  • This section of the fuzzy logic describes
  • Basic Definition of Fuzzy Set
  • Similarities and Differences of Fuzzy Sets with
    Traditional Set Theory
  • Examples Illustrating the Concepts of Fuzzy Sets
  • Logical Operation on Fuzzy Sets
  • Hedging

17
Fuzzy Set
  • A fuzzy set is a set whose elements have degrees
    of membership.
  • That is, a member of a set can be full member
    (100 membership status) or a partial member (eg.
    less than 100 membership and greater than 0
    membership).
  • To fully understand fuzzy sets, one must first
    understand traditional sets.
  • A traditional or crisp set can formally be
    defined as the following
  • A subset U of a set S is a mapping from the
    elements of S to the elements of the set 0,1.
    This is represented by the notation U S-gt
    0,1
  • The mapping is represented by one ordered pair
    for each element S where the first element is
    from the set S and the second element is from the
    set 0,1.
  • The value zero represents non-membership, while
    the value one represents membership.
  • Essentially this says that an element of the set
    S is either a member or a non-member of the
    subset U. There are no partial members in
    traditional sets.

18
  • A fuzzy set is a set whose elements have degrees
    of membership.
  • These can formally be defined as the following
  • A fuzzy subset F of a set S can be defined as a
    set of ordered pairs. The first element of the
    ordered pair is from the set S, and the second
    element from the ordered pair is from the
    interval 0,1.
  • The value zero is used to represent
    non-membership the value one is used to
    represent complete membership, and the values in
    between are used to represent degrees of
    membership.

19
Example
  • Consider a set of young people using fuzzy sets.
  • In general, young people range from the age of 0
    to 20.
  • But, if we use this strict interval to define
    young people, then a person on his 20th birthday
    is still young (still a member of the set). But
    on the day after his 20th birthday, this person
    is now old (not a member of the young set).
  • How can one remedy this?
  • By RELAXING the boundary between the strict
    separation of young and old.
  • This separation can easily be relaxed by
    considering the boundary between young and old as
    "fuzzy".
  • The figure below graphically illustrates a fuzzy
    set of young and old people.

20
  • Notice in the figure that people whose ages are
    gt zero and lt 20 are
  • complete members of the young set (that is, they
    have a membership value of one).
  • Also note that people whose ages are gt 20 and lt
    30 are partial members of
  • the young set.
  • For example, a person who is 25 would be young to
    the degree of 0.5.
  • Finally people whose ages are gt 30 are
    non-members of the young set.

21
Logical Operations on Fuzzy Sets
  • Negation
  • Intersection
  • Union

22
Negation
  • the red line is a fuzzy set.
  • To negate this fuzzy set, subtract the membership
    value in the fuzzy set from 1.
  • For example, the membership value at 5 is one. In
    the negation, the membership value at 5 would be
    (1-10) and if the membership value is 0.4 the
    membership value would be (1-0.40.6).

23
Intersection
  • In this figure, the red and green lines are fuzzy
    sets.
  • To find the intersection of these sets take the
    minimum of the two membership values at each
    point on the x-axis.
  • For example, in the figure the red fuzzy set has
    a membership of ZERO when x 4 and the green
    fuzzy set has a membership of ONE when x 4.
  • The intersection would have a membership value of
    ZERO when x 4 because the minimum of zero and
    one is zero.

24
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25
Union
  • To find the union of these sets take the maximum
    of the two membership values at each point on the
    x-axis.
  • For example, in the figure the red fuzzy set has
    a membership of ZERO when x 4 and the green
    fuzzy set has a membership of ONE when x 4. The
    union would have a membership value of ONE when x
    4 because the maximum of zero and one is One.

26
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27
Limitation
  • Fuzzy logic cannot be used for unsolvable
    problems.
  • An obvious drawback to fuzzy logic is that it's
    not always accurate. The results are perceived as
    a guess, so it may not be as widely trusted as an
    answer from classical logic. Certainly, though,
    some chances need to be taken. How else can
    dressmakers succeed in business by assuming the
    average height for women is 5'6"?
  • Fuzzy logic can be easily confused with
    probability theory, and the terms used
    interchangeably. While they are similar concepts,
    they do not say the same things.
  • Probability is the likelihood that something is
    true. Fuzzy logic is the degree to which
    something is true (or within a membership set).

28
  • Classical logicians argue that fuzzy logic is
    unnecessary. Anything that fuzzy logic is used
    for can be easily explained using classic logic.
    For example, True and False are discrete. Fuzzy
    logic claims that there can be a gray area
    between true and false.
  • Fuzzy logic has traditionally low respectability.
    That is probably its biggest problem. While fuzzy
    logic may be the superset of all logic, people
    don't believe it. Classical logic is much easier
    to agree with because it delivers precision.

29
References
  • http//www.dementia.org/julied/logic/index.html
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