Reasoning with Imprecision

• Fuzzy Sets [Lug Ch.8.2.2]
  • The items in a fuzzy set have a membership function to represent the degree of membership of each item.
  • Example:
    • tall = {0 if height<5, (height-5)/2 if 5<=height<=7, 1 if height>7}
    • medium = {0 if height<3, height-3 if 3<=height<=4, 1 if 4<=height<5, 6-height if 5<=height<=6, 0 if height>6}
    • short = {1 if height<2, (4-height)/2 if 2<=height<=4, 0 if height>4}
    • A graph of this looks like:

           Short    Medium    Tall
      1.0 +----+     +--+     +-------------------
          |     \    |  |    /
          |      \  /    \  /
      0.5 +       \|      |/
          |       /\      /\
          |       | \    / |
      0.0 +-+--+--+--+--+--+--+-------------------
            |     |     |     |
           1.0   3.0   5.0   7.0
                     

    • Here are some example values:

      Height    shortness mediumness tallness
      --------------------------------------
      1' 2"     1.00      0.00       0.00
      2' 6"     0.75      0.00       0.00
      3' 2"     0.42      0.16       0.00
      4' 5"     0.00      1.00       0.00
      5' 9"     0.00      0.25       0.38
      6' 1"     0.00      0.00       0.54
      7' 2"     0.00      0.00       1.00
                     

  • Expressions like "A is X" can be interpreted as degrees of truth, e.g., "5'9" is tall" = 0.38.
  • Membership functions used in most applications almost never have such simple shapes
  • Also (occasionally), may have multiple criteria: tall depends on both a person's height and their age (he's tall for his age). A two-dimensional membership function, or a "fuzzy relation"
  • Logic operations
    • truth(~X) = 1.0 - truth(X)
    • truth(X & Y) = minimum(truth(X), truth(Y))
    • truth(X | Y) = maximum(truth(X), truth(Y))
  • Examples
    • Definition of tall as above
    • old = {0 if age<18, (age-18)/42 if 18<=age<=60, 1 if age > 60}
    • Here are some example values:

      Height Age  tall  old  tall & old tall | old ~tall
      ------------------------------------------------
      1' 2"   12  0.00  0.00    0.00       0.00    1.00
      3' 2"   50  0.00  0.76    0.00       0.76    1.00
      5' 9"   65  0.38  1.00    0.38       1.00    0.62
      7' 2"   30  1.00  0.28    0.28       1.00    0.00
                     

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