Artificial Intelligence CIS 342

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Artificial IntelligenceCIS 342

• The College of Saint Rose
• David Goldschmidt, Ph.D.

March 20, 2008
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Fuzzy Logic

• Unlike Boolean logic, fuzzy logic is multi-valued
  
• Fuzzy logic represents degrees of membershipand
  degrees of truth
• Things can be part true and part false at the
  same time

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Linguistic Variables Hedges

• Hedges are qualifying terms that modifythe shape
  of fuzzy sets

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Fuzzy Set Operations

• Complement
• To what degree do elements not belong to this set?

mA(x) 1 mA(x)
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Fuzzy Set Operations

• Containment
• Which sets belong to other sets?

Each element of the fuzzy subset has smaller
membership than in the containing set
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Fuzzy Set Operations

• Intersection
• To what degree is the element in both sets?

mAnB(x) min mA(x), mB(x)
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Fuzzy Set Operations

• Union
• To what degree is the element in either or both
  sets?

mA?B(x) max mA(x), mB(x)
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Fuzzy Rules

• 1965 paper Fuzzy Sets (Lotfi Zadeh)
• Apply natural language terms to a formal
  system of mathematical logic
• http//www.cs.berkeley.edu/zadeh
• 1973 paper outlined a new approach to capturing
  human knowledge and designing expert systems
  using fuzzy rules

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Fuzzy Rules

• A fuzzy rule is a conditional statementin the
  familiar form
• IF x is A
• THEN y is B
• where x and y are linguistic variables, and A and
  B are linguistic values determined by fuzzy sets
  on the universe of discourses X and Y,
  respectively

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Linguistic Variables

• A linguistic variable is a fuzzy variable
• e.g. the fact John is tall implies linguistic
  variable John takes the linguistic value
  tall
• Use linguistic variables to form fuzzy rules

IF project duration is long THEN risk is
high IF speed is slow THEN stopping distance
is short
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Fuzzy Expert Systems

• A fuzzy expert system is an expert system
  thatuses fuzzy rules, fuzzy logic, and fuzzy
  sets
• Many (or all) rules in a fuzzy logic system fire
  to some extent
• If the antecedent is true to some degree of
  membership, then the consequent is true to the
  same degree

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Fuzzy Expert Systems

• Two distinct fuzzy sets describing tall and heavy

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Fuzzy Expert Systems

• IF height is tall
• THEN weight is heavy

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Fuzzy Expert Systems

• Other examples (multiple antecedents)
• e.g. IF project duration is long
• AND project staffing is large
• AND project funding is inadequate
• THEN risk is high
• e.g. IF service is excellent
• OR food is delicious
• THEN tip is generous

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Fuzzy Expert Systems

• Other examples (multiple consequents)
• e.g. IF temperature is hot
• THEN hot water is reduced
• cold water is increased

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Fuzzy Inference

• Aptly named after Ebrahim Mamdani, theMamdani
  method for fuzzy inference is
• 1. Fuzzification of the input variables
• 2. Rule evaluation
• 3. Aggregation of the rule outputs
• 4. Defuzzification

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Fuzzy Inference Example

• Rule 1IF x is A3OR y is B1THEN z is C1
• Rule 2IF x is A2AND y is B2THEN z is C2
• Rule 3IF x is A1THEN z is C3

• Rule 1IF project funding is
  adequateOR project staffing is
  smallTHEN risk is low
• Rule 2IF project funding is
  marginalAND project staffing is
  largeTHEN risk is normal
• Rule 3IF project funding is
  inadequateTHEN risk is high

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Fuzzy Inference Example

• 1. Fuzzification

project funding
project staffing
inadequate
small
marginal
large
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Fuzzy Inference Example

• 2. Rule 1 evaluation

risk
project staffing
project funding
adequate
small
low
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Fuzzy Inference Example

• 2. Rule 2 evaluation

project funding
project staffing
risk
marginal
large
normal
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Fuzzy Inference Example

• 2. Rule 3 evaluation

risk
project funding
inadequate
high
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Fuzzy Inference Example

• 3. Aggregation of the rule outputs

risk
high
normal
low
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Fuzzy Inference Example

• 4. Defuzzification
• e.g. use the centroid method in which a vertical
  line slices the aggregate set into two equal
  halves
• How can we calculate this?

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Fuzzy Inference Example

• 4. Defuzzification
• Calculate the centre of gravity (cog)

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Fuzzy Inference Example

• 4. Defuzzification
• Use a reasonable sampling of points

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Applications of Fuzzy Logic

• Why use fuzzy expert systems or fuzzy control
  systems?
• Apply fuzziness (and therefore accuracy) to
  linguistically defined terms and rules
• Lack of crisp or concrete mathematical models
  exist
• When do you avoid fuzzy expert systems?
• Traditional approaches produce acceptable results
• Crisp or concrete mathematical models exist and
  are easily implemented

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Applications of Fuzzy Logic

• Real-world applications include
• Control of robots, engines, automobiles,
  elevators, etc.
• Cruise-control in automobiles
• Temperature control
• Reduce vibrations in camcorders
• http//www.esru.strath.ac.uk/Reference/concepts/fu
  zzy/fuzzy_appl.de20.htm
• Handwriting recognition, OCR
• Predictive and diagnostic systems (e.g. cancer)