Fuzzy Logic

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Fuzzy Logic
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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.

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Fuzziness Vs. Vagueness
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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
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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
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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
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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.

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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.

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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.

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• 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.

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Fuzzy Logic
Boolean Logic
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• 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.

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• 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.

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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.

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• 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.

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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

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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.

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• 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.

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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.

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• 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.

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Logical Operations on Fuzzy Sets

• Negation
• Intersection
• Union

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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).

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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.

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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.

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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).

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• 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.

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References

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