2. Introduction to fuzzy logic

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					 Fuzzy Sets and Fuzzy Logic
              Fuzzy Graphs
  Fuzzy Associative Memory




2. Introduction to fuzzy logic

              Philippe De Wilde

         Department of Computer Science
             Heriot-Watt University
                   Edinburgh


       Vloeberghs Leerstoel 2010




          Philippe De Wilde   2. Introduction to fuzzy logic
                Fuzzy Sets and Fuzzy Logic
                             Fuzzy Graphs
                 Fuzzy Associative Memory


Outline



  1   Fuzzy Sets and Fuzzy Logic


  2   Fuzzy Graphs


  3   Fuzzy Associative Memory




                         Philippe De Wilde   2. Introduction to fuzzy logic
               Fuzzy Sets and Fuzzy Logic
                            Fuzzy Graphs
                Fuzzy Associative Memory


Six Lectures


     1. History of decision making under uncertainty. Game:
     repeated prisoner’s dilemma.
     2. Introduction to fuzzy logic. Game: rock, scissors, paper.
     3. Financial markets and fuzzy games. Game: fuzzy
     chess.
     4. Coupled networks in the brain. Game: Monty Hall’s
     dilemma.
     5. Neuroeconomics. Game: ultimatum game.
     6. The AI of deviations of rationality.



                        Philippe De Wilde   2. Introduction to fuzzy logic
             Fuzzy Sets and Fuzzy Logic
                          Fuzzy Graphs
              Fuzzy Associative Memory


Linguistic Variables




                      Philippe De Wilde   2. Introduction to fuzzy logic
                   Fuzzy Sets and Fuzzy Logic
                                Fuzzy Graphs
                    Fuzzy Associative Memory


Membership Function

  membership

  1


               fast



  0
       30 60 90 120 speed

  µfast (speed),    µA (x).


                            Philippe De Wilde   2. Introduction to fuzzy logic
                  Fuzzy Sets and Fuzzy Logic
                               Fuzzy Graphs
                   Fuzzy Associative Memory


Membership Function Definition

  Definition (Universe of discourse)
  The universe of discourse is a set X , discrete ({x1 , . . . , xn }), or
  continuous (union of intervals on the real line).

  Definition (Membership Function)
  A membership function is a function µA : X → [0, 1].

  Definition (Fuzzy Set)
  A fuzzy set is defined by a membership function, it consists of
  some elements x of a universe of discourse X together with
  their membership values (or degrees) µa (x).


                           Philippe De Wilde   2. Introduction to fuzzy logic
                 Fuzzy Sets and Fuzzy Logic
                              Fuzzy Graphs
                  Fuzzy Associative Memory


Basic Logic Operations


  Definition (AND)
  µA∩B (x) = min(µA (x), µB (x)),             ∀x ∈ X .

  Definition (OR)
  µA∪B (x) = max(µA (x), µB (x)),             ∀x ∈ X .

  Definition (NOT, optional)
  µ¬A (x) = 1 − µA (x),       ∀x ∈ X .




                          Philippe De Wilde    2. Introduction to fuzzy logic
                Fuzzy Sets and Fuzzy Logic
                             Fuzzy Graphs
                 Fuzzy Associative Memory


Extension Principle


  A function transforming a set into another set will transform a
  membership function into another membership function, using
  the extension principle.
  Definition (Extension Principle)
  If f : X → Y is a function transforming universe of discourse X
  into Y , then fuzzy set µA (x) is transformed into µB (y ):

                           maxy =f (x) µa (x) if f −1 (y ) = ∅,
           µB (y ) =
                           0                  otherwise.




                         Philippe De Wilde   2. Introduction to fuzzy logic
                Fuzzy Sets and Fuzzy Logic
                             Fuzzy Graphs
                 Fuzzy Associative Memory


Example of Extension Principle


  y = x2




  The extension principle is powerful, and can be used to create
  fuzzy arithmetic.



                         Philippe De Wilde   2. Introduction to fuzzy logic
           Fuzzy Sets and Fuzzy Logic
                        Fuzzy Graphs
            Fuzzy Associative Memory


Fuzzy Numbers
 

 

 

 

 
                             µ10(x) 
 

 

 

 

 

                                                                   x
    0 
                      10



                    Philippe De Wilde   2. Introduction to fuzzy logic
                 Fuzzy Sets and Fuzzy Logic
                              Fuzzy Graphs
                  Fuzzy Associative Memory


Decision Making under Fuzziness

  Definition
  Let X be a set of options. A fuzzy goal is a fuzzy set
  µG (x), x ∈ X . A fuzzy constraint is a fuzzy set µC (x), x ∈ X .

  A fuzzy decision is a fuzzy set µD (x), x ∈ X , with

                      µD (x) = min(µG (x), µC (x)).


  A crisp decision x ∗ can be derived from a fuzzy decision by
  defuzzification:
                         x ∗ = arg max µD (x).
                                              x∈X

  There are several ways to defuzzify, for example the centre of
  gravity of the area under the curve.
                          Philippe De Wilde     2. Introduction to fuzzy logic
                 Fuzzy Sets and Fuzzy Logic
                              Fuzzy Graphs
                  Fuzzy Associative Memory


Example of Decision Making



               µG(x)
                                                  µC(x)




                                                                  x
                                 X*
  x ∗ is the decision, subject to constraints C and goal G.



                          Philippe De Wilde   2. Introduction to fuzzy logic
               Fuzzy Sets and Fuzzy Logic
                            Fuzzy Graphs
                Fuzzy Associative Memory


Uncertainty of a Functional Dependency
     Uncertainty can be represented by additive noise, e.g.
                                      y = x 2 + ξ,
     with ξ a random variable.
     The noise can also be on the parameters of the functional
     relationship, e.g.
                             y = x (2+ξ) ,
     or
                                        y = ξx 2 .

     Zadeh proposed a radically different way of looking at this,
     where the curve of a function becomes a union of squares,
     and each point in the union belongs to the function to a
     certain degree. This is the fuzzy graph.
                        Philippe De Wilde   2. Introduction to fuzzy logic
              Fuzzy Sets and Fuzzy Logic
                           Fuzzy Graphs
               Fuzzy Associative Memory


Fuzzy Graph




                       Philippe De Wilde   2. Introduction to fuzzy logic
                  Fuzzy Sets and Fuzzy Logic
                               Fuzzy Graphs
                   Fuzzy Associative Memory


Union of Cartesian Products


  The fuzzy graph is a union: (A1 × B1 ) ∪ (A2 × B2 ) ∪ . . . (An × Bn ).
  Definition (Fuzzy graph)
  If X and Y are universes of discourse, f ∗ : X → Y is a fuzzy
  graph iff
               n
       f∗ =    i=1 Ai   × Bi ,
       µf ∗ (u, v ) = maxi min(µAi (u), µBi (v )),                u ∈ X, v ∈ Y.

  max and min come from (u is in A1 AND v is in B1 ), OR (u is in
  A2 AND v is in B2 ), OR ...




                           Philippe De Wilde   2. Introduction to fuzzy logic
                Fuzzy Sets and Fuzzy Logic
                             Fuzzy Graphs
                 Fuzzy Associative Memory


Fuzzy Graph as a Set of Rules


     e.g. f ∗ represents:
     If u is small then v is large
     else (or) if u is medium then v is medium
     else if u is large then v is small.
     If u is A1 then v is B1
     else if u is A2 then v is B2
     else if u is A3 then v is B3 .
     Bi can coincide with Bj , but the Ai need to be different, just
     as f(x) needs to be unique for a function.
     Fuzzy probability distributions are possible!



                         Philippe De Wilde   2. Introduction to fuzzy logic
                Fuzzy Sets and Fuzzy Logic
                             Fuzzy Graphs
                 Fuzzy Associative Memory


Intersection of Two Fuzzy Graphs




  Equivalent of solving two simultaneous equations.


                         Philippe De Wilde   2. Introduction to fuzzy logic
                Fuzzy Sets and Fuzzy Logic
                             Fuzzy Graphs
                 Fuzzy Associative Memory


Inversion of a Fuzzy Graph




  If a fuzzy set B is given, find the fuzzy set A corresponding with
  it according to f ∗ .
  Minimum and maximum of a fuzzy graph can also be defined.

                         Philippe De Wilde   2. Introduction to fuzzy logic
               Fuzzy Sets and Fuzzy Logic
                            Fuzzy Graphs
                Fuzzy Associative Memory


Fuzzy Control: Rules for Stopping a Car


     1 input, 1 rule, 1 output
     If you go too fast, brake hard.
     1 input, 2 rules, 1 output
     If you go too fast, brake hard, or, if you go fast, brake.
     2 inputs, 4 rules, 1 output
     If you go too fast and the wall is very close, brake hard, or
     If you go fast and the wall is very close, brake, or
     If you go too fast and the wall is close, brake, or
     If you go fast and the wall is close, slow down.




                        Philippe De Wilde   2. Introduction to fuzzy logic
               Fuzzy Sets and Fuzzy Logic
                            Fuzzy Graphs
                Fuzzy Associative Memory


Fuzzy Associative Memory


                     too fast               fast
      very close     brake hard             brake
      close          brake                  slow down
           µ2 µ2 µ 2 . . .
             1    2    3
      µ1
       1   µ11 µ12 µ13 . . .
      µ1
       2   µ21 µ22 µ23 . . .
      µ1
       3   µ31 µ32 µ33 . . .
      .
      .     .
            .    .
                 .    .
                      .   ..
      .     .    .    .      .
     That’s how an expert lists the rules she uses.




                        Philippe De Wilde    2. Introduction to fuzzy logic
                Fuzzy Sets and Fuzzy Logic
                             Fuzzy Graphs
                 Fuzzy Associative Memory


2 Inputs Fire Rules


     2 inputs x 1 and x 2
     x 1 belongs to the input membership functions µ1 , µ1 , µ1 , . . .
                                                        1 2 3
     to degrees µ1 (x 1 ), µ1 (x 1 ), µ1 (x 1 ), . . ..
                  1         2          3
     x 2 belongs to the input membership functions µ2 , µ2 , µ2 , . . .
                                                        1 2 3
     to degrees µ2 (x 2 ), µ2 (x 2 ), µ2 (x 2 ), . . ..
                  1         2          3
     output membership function µij fires at degree
     min[µ1 (x 1 ), µ2 (x 2 )], using min because of the ’and’ in the
            i        j
     rules.
     output membership function µij is truncated at
     min[µ1 (x 1 ), µ2 (x 2 )].
          i          j




                         Philippe De Wilde   2. Introduction to fuzzy logic
                              Fuzzy Sets and Fuzzy Logic
                                           Fuzzy Graphs
                               Fuzzy Associative Memory
 
Truncated Output Membership Function
 
 
                                                  µij(z) 
                                                                
                                                                     min[µi1(x1),µj2(x2)] 
 

                                                                                                z

            truncation
            = min µij (z), min[µ1 (x 1 ), µ2 (x 2 )]
                                i          j

            = min µij (z), µ1 (x 1 ), µ2 (x 2 )
                            i          j

                                        Philippe De Wilde      2. Introduction to fuzzy logic
                     Fuzzy Sets and Fuzzy Logic
                                  Fuzzy Graphs
                      Fuzzy Associative Memory


    Combination of all Output Membership Functions
         Max, because a collection of rules is combined with ’or’.
         maxi,j min µij (z), µ1 (x 1 ), µ2 (x 2 )
                              i          j
         Sum can be used instead of max.




                                                                              z
                              Philippe De Wilde   2. Introduction to fuzzy logic
                 Fuzzy Sets and Fuzzy Logic
                              Fuzzy Graphs
                  Fuzzy Associative Memory


Defuzzification using Centre of Gravity




     f (z) = maxi,j min µij (z), µ1 (x 1 ), µ2 (x 2 )
                                  i          j
                                      ∞
                                      −∞ zf (z)dz
     Centre of gravity y =             ∞
                                      −∞ f (z)dz

     y is the defuzzified output, the control




                          Philippe De Wilde   2. Introduction to fuzzy logic
         Fuzzy Sets and Fuzzy Logic
                      Fuzzy Graphs
          Fuzzy Associative Memory




Applications in automotive, process control, consumer
electronics, defence
Abe Mamdani




                  Philippe De Wilde   2. Introduction to fuzzy logic
             Fuzzy Sets and Fuzzy Logic
                          Fuzzy Graphs
              Fuzzy Associative Memory


Summary


    Membership functions define fuzzy sets.
    Fuzzy graphs express relationships between linguistic
    variables, a collection of fuzzy if-then-else rules.
    Fuzzy inference uses fuzzy associative memory, and
    defuzzification.


    Next
        game theory
        financial markets
        fuzzy games



                      Philippe De Wilde   2. Introduction to fuzzy logic
                            Appendix    For Further Reading



Further Reading and Picture Credits I




     Lotfi Zadeh
     Fuzzy Logic, Neural Networks, and Soft Computing.
     Communications of the ACM, 37(3):77–84, 1994.




                    Philippe De Wilde   2. Introduction to fuzzy logic