Fuzzy Logic

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Fuzzy Logic
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Fuzzy Logic

• Fuzzy logic can be viewed as an extension of
  multi-valued logic.
• Fuzzy logic deals with the approximate rather
  than precise models.
• Fuzzy logic is a matter of degree.

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Basic Differences Between Two Logics

• In two-valued logic systems, a proposition p is
  either true or false.
• In fuzzy logic, the truth values are allowed to
  range over the fuzzy subsets of a finite or
  infinite truth value set T.
• The predicates in two-valued logic are
  constrained to be crisp.
• In fuzzy logic, the predicates may be crisp
• E.g., mortal, even, etc.
• They can also be more general
• E.g., ill, tired, tall, very tall, etc.

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Basic Difference Between Two Logics (Contd)

• Two-valued logic allows only two quantifiers
  all and some.
• In fuzzy logic, it allows, in addition, theuse of
  fuzzy quantifiers most, few, many,
  several, much of, etc.
• In two-valued logical systems, a p may be
  quantified by associating with p
• Truth value, true or false
• A modal operator such as possible or
  necessary
• An intensional operator such as know or
  believe

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Basic Difference Between Two Logics (Contd)

• Fuzzy logic has three principal modes of
  qualification
• Truth-qualification, as in (Mary is young) is not
  quite true.
• Probability-qualification, as in (Mary is young)
  in unlikely.
• Possibility-qualification, s in (Mary is young)
  is almost impossible.

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Meaning Representation and Inference
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Canonical Form
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Canonical Form (Example)
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Inference Rules

• Categorical
• Rules that do not contain fuzzy quantifiers
• Dispositional
• Rules in which one or more premises may contain,
  explicitly or implicitly, the fuzzy quantifier
  usually.

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Inference Rules (Contd)
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Inference Rules (Contd)
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Inference Rules (Contd)
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Linguistic Variable

• Definition A linguistic variable is a variable
  whose values are words or sentences in a natural
  or synthetic language.
• For example
• Age is a linguistic variable if its values are
  young, anot young, and so on.
• In general, the values of the linguistic variable
  can be generated from a primary term (for
  example, young), its antonym (old), a
  collection of modifiers (not, very, quite),
  and the connectives and and or.
• Furthermore, each value represents a possibility
  distribution

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Fuzzy Car by Sugeno
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Control Rules
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Fuzzy Controllers

• Fuzzy controllers are modeled according to the
  human behavior.
• Fuzzy controllers are simpler, since they have a
  smaller number of rules.
• Trade-off between imprecision and simplification.