Fuzzy Sets and Fuzzy Logic

1
Fuzzy Sets and Fuzzy Logic

• CISC 204
• Tony Kuo

2
Brief History

• Lukasiewicz
• in 1920 developed the first 3 valued logic system
• experimented with 4 valued logic and hypothesized
  about infinite valued logic
• Zadeh
• in 1965 developed the first infinite valued logic
  in the form of fuzzy sets
• Research into fuzzy logic is ongoing
• notably in combination with other computing
  techniques
• Many commercial applications exist
• control theory is the most popular application
  domain
• real-world scenarios which are inherently vague

3
Membership

• The idea of membership is the core concept of
  fuzzy logic
• Every object in the universe has a degree of
  membership to some set
• The tea is hot
• object tea is a member of hot
• where tea is an object in the universe
• where hot is a set of descriptors for tea
• symbolically function m(HOT) returns a value
  01 representing the degree of membership of
  tea to hot

4
Fuzzy Sets

• A fuzzy set must satisfy the following criteria
• there exists a universe of discourse
• every object in the universe is a member of a
  fuzzy set
• the fuzzy set can be in the universe of discourse
  or in another universe
• For example
• in the universe A 1, 2, 3
• A(1) 1, 0.8, 0.2
• A(2) 0.8, 1, 0.8
• A(3) 0.2, 0.8, 1
• in laymens terms A(1) represents A is 1
• A is 1 has a truth value of 1, A is 2 has a
  truth value of 0.8, etc.

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Visual Example

• typically, membership is represented in graph
  form
• universe of objects is people
• universe of fuzzy set is height
• fuzzy set of a person at 6 tall is thus
• short, medium, tall 0.25, 0.5, 0

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Ways to determine membership functions

• Intuition, first principles
• applications that involve known mechanisms
  physics, biology, mechanical
• Heuristics neural networks, genetic algorithms,
  etc.
• applications that are data driven and the first
  principles are unknown or hard
• Ranking, voting
• applications involving human behaviour, such as
  social phenomenon, culture

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Ranking Exercise

• Rank the following in descending order of
  membership to fish
• Crocodile
• Whale
• Tuna
• Eel
• Frog
• Snake
• The averaged end result of this survey is a fuzzy
  set of membership to fish
• This technique is best done with experts opinion

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Implications

• Traditional (Aristotle) logic
• All propositions are either true or false
• P or ØP true (law of excluded middle), modus
  ponins, modus tonins, proof by contradictions,
  etc.
• Canon of knowledge is well defined
• However, fuzzy x implies y has many
  definitions, all of which are correct
• standard strict 1 if x y, else 0
• Mamdani min (x, y)
• Lukasiewicz min (1, 1-xy)
• Larson xy
• Godel 1 if x y, else y

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Fuzzy Logic Operators

• There exists many different definitions of fuzzy
  logic operators, but the following are the most
  common.
• A or B max (A, B)
• A and B min (A, B)
• ØA 1 A
• A Í B A and B / A, measure of subsethood

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Generalized Modus Ponins

• if x is A, then y is B
• x is A
• ------------------------
• then y is B
• y is B needs to be inferred from the premise A
  B
• How do we define B ?
• max min composition
• B max min(A, A B)
• Note if there are multiple premises, they are
  connected by and

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Example

• First we calculate
• A B
• Then
• B(4) max 0.3, 0.8, 1
• B(5) max 0.3, 0.8, 1
• B(6) max 0.3, 0.3, 0.3
• B(7) max 0, 0, 0
• B 1, 1, 0.3, 0

• Given universe
• U 1, 2, 3
• V 4, 5, 6, 7
• Given fuzzy sets
• A u is 2 0.8, 1, 0.8
• B v is 4 1, 0.8, 0.3, 0
• Given Godel implication
• A B 1 if A B, else B
• Given A
• u is 3 0.3, 0.8, 1

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• B 1, 1, 0.3, 0
• this result is a fuzzy set, and not a definite
  crisp answer
• a defuzzified answer can be found in various ways
• a popular way is to find the centre of mass
• thus we can say
• if u 2 then v 4
• u 3
• -------------------------------
• then v 5.42
• there exists other methods to defuzzifiy fuzzy
  sets

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A Fuzzy Control System

• A fuzzy control system requires the following
• inputs
• outputs
• descriptors for the inputs and outputs (status)
• membership functions to relate inputs and outputs
  to their statuses

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Example Fuzzy Control System

• Temperature Control
• 2 inputs
• current temperature, rate of change of
  temperature
• 3 possible outputs
• call for cooling, dont change, call for heating
• 3 descriptors for current temperature
• too cold, just right, too hot
• 3 descriptors for rate of change of temperature
• getting colder, no change, getting hotter

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• We need to determine a control systems
  objectives
• This is usually done either by examining data or
  consulting an expert
• This is typically written as if x and y then z
  rules
• We need membership functions for each input and
  output for all descriptors

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Simulation / Inference

• Simulation and inference on this system can be
  done using the generalized modus ponins paradigm
  presented earlier
• if x is A1 and y is B1 then z is C1
• ¼
• if x is An and y is Bn then z is Cn
• x is A and y is B
• ------------------------------------
• z is C

A is in the universe of temperature B is in the
universe of change of temperature C is in the
universe of output
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• Get input x0 is A and y0 is B, ie. measure the
  temperature and rate of change of temperature
• Determine Ai and Bi Ci for x0 for each i using
  max min composition
• Determine C as a combination of Ci
• Determine the specific output (defuzzification)
• Combining Ci can be done as a union,
  intersection, or various other aggregation
  operators

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System Rules Simplifies Inference

• temperature -1
• too cold, just right, too hot
• 0.5, 0.5, 0
• change in temperature 2.5
• getting colder, no change, getting hotter
• 0, 0.5, 0.5
• just plug in the appropriate truth values into
  the rules and determine which rules fired
• defuzzify this output
• however, system rules arent always known and are
  perhaps what were trying to reverse engineer