Artificial Intelligence CIS 342

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Artificial IntelligenceCIS 342

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

March 14, 2008
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Fuzzy Logic

• Expert knowledge often uses vague and inexact
  terms
• Fuzzy Logic describes fuzziness by specifying
  degrees
• e.g. degrees of height, speed, distance,
  temperature, beauty, intelligence, etc.

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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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Fuzzy Sets

• A comparison of crisp and fuzzy sets depicting
  height

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Fuzzy Sets

• A crisp (or Boolean) set is too sharp
• Low applicability to real-world knowledge/concepts

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Fuzzy Sets

• X-axis is the universe of discourse, all possible
  values
• Y-axis is the degree of membership

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Representing Fuzzy Sets

• Representing height using three fuzzy sets

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In-Class Exercise
write a function or method tocalculate degree of
membership (HINT use analytic geometry)

• Determine the degree ofmembership in each fuzzy
  set

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Qualifiers Hedges

• What about linguistic values with qualifiers?
• e.g. very tall, extremely short, etc.
• Hedges are qualifying terms that modifythe shape
  of fuzzy sets
• e.g. very, somewhat, quite, slightly, extremely,
  etc.

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Representing Hedges
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Representing Hedges
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Representing Hedges
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Representing Hedges
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Linguistic Variables Hedges
write a function or method called very()
thatmodifies the degree of membershipe.g.
double x very( tall( 185 ) )
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Crisp Set Operations

• Crisp set operations developed by Georg Cantor in
  the late 19th century

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Crisp Set Operations

• Crisp set operations developed by Georg Cantor in
  the late 19th century (continued)

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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)