Models for Inexact Reasoning Fuzzy Logic – Lesson 1...

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  • Models for Inexact Reasoning

    Fuzzy Logic Lesson 1Crisp and Fuzzy Sets

    Master in Computational Logic

    Department of Artificial Intelligence

  • Origins and Evolution of Fuzzy Logic

    Origin: Fuzzy Sets Theory (Zadeh, 1965)

    Aim: Represent vagueness and impre-cission of statements in natural language

    Fuzzy sets: Generalization of classical (crisp) sets

    In the 70s: From FST to Fuzzy Logic

    Nowadays: Applications to control systems

    Industrial applications

    Domotic applications, etc.

  • Fuzzy Logic

    Fuzzy Logic - Lotfi A. Zadeh, Berkeley

    Superset of conventional (Boolean) logic that has been extended to handle the concept of partial truth

    Truth values (in fuzzy logic) or membership values (in fuzzy sets) belong to the range [0, 1], with 0 being absolute Falseness and 1 being absolute Truth.

    Deals with real world vagueness

  • Real-World Applications

    ABS Brakes

    Expert Systems

    Control Units

    Bullet train between Tokyo and Osaka

    Video Cameras

    Automatic Transmissions

  • Crisp (Classic) Sets

    Classic subsets are defined by crisp predicates

    Crisp predicates classify all individuals into two groups or categories

    Group 1: Individuals that make true the predicate

    Group 2: Individuals that make false the predicate

    Example:

    Predicate: n is odd{ }| 1 2 ,E

    A E n E n k k Z

    =

    = = +

    Z

  • Crisp Characteristic Functions

    The classification of individuals can be done using a indicator or characteristic function:

    Note that:

    { }: 0,1

    1,( )

    0,

    A

    A

    E

    x Ax

    x A

    =

    { }

    { }

    1

    1

    (1) , 3, 1,1,3,

    (0) , 4, 2,0, 2,4,

    A

    A

    =

    =

    K K

    K K

  • Fuzzy Sets

    Human reasoning often uses vague predicates Individuals cannot be classified into two groups!

    (either true or false)

    Example: The set of tall men But what is tall?

    Height is all relative

    As a descriptive term, tall is very subjective and relies on the context in which it is used

    Even a 5ft7 man can be considered "tall" when he is surrounded by people shorter than he is

  • Fuzzy Membership Functions

    It is impossible to give a classic definition for the subset of tall men

    However, we could establish to which degree a man can be considered tall

    This can be done using membership functions:

    : [0,1]A E

  • Fuzzy Membership Functions

    A(x) = y

    Individual x belongs to some extent (y) to subset A

    y is the degree to which the individual x is tall

    A(x) = 0

    Individual x does not belong to subset A

    A(x) = 1

    Individual x definitelly belongs to subset A

  • Types of Membership Functions

    Gaussian

  • Types of Membership Functions

    Triangular

  • Types of Membership Functions

    Trapezoidal

  • Example

    E = {0, , 100} (Age)

    Fuzzy sets: Young, Mature, Old

  • Membership Functions

    Membership functions represent distributions of possibility rather than probability

    For instance, the fuzzy set Young expresses the possibility that a given individual be young

    Membership functions often overlap with each others A given individual may belong to different fuzzy sets

    (with different degrees)

  • Membership Functions For practical reasons, in many cases the

    universe of discourse (E) is assumed to be discrete

    { }1 2, , , nE x x x= K

    The pair (A(x), x), denoted by A(x)/x is called fuzzy singleton

    Fuzzy sets can be described in terms of fuzzy singletons

    { }1

    ( ( ) / ) ( ) /n

    A A i ii

    A x x x x =

    = = U

  • Basic Definitions over Fuzzy Sets

    Empty set: A fuzzy subset A E is empty (denoted A = ) iff

    ( ) 0,A x x E =

    Equality: two fuzzy subsets A and B defined over E are equivalent iff

    ( ) ( ),A Bx x x E =

  • Basic Definitions over Fuzzy Sets

    A fuzzy subset A E is contained in B E iff

    ( ) ( ),A Bx x x E

    Normality: A fuzzy subset A E is said to be normal iff

    max ( ) 1Ax E

    x

    =

    Support: The support of a fuzzy subset A E is a crisp set defined as follows

    { }| ( ) 0A A AS x E x S E = >

  • Operations over Fuzzy Sets

    The basic operations over crisp sets can be extended to suit fuzzy sets

    Standard operations:

    Intersection:

    Union:

    Complement:

    ( ) min( ( ), ( ))A B A Bx x x =

    ( ) max( ( ), ( ))A B A Bx x x =

    ( ) 1 ( )AA x x =

  • Operations over Fuzzy Sets

    Intersection

  • Operations over Fuzzy Sets

    Union

  • Operations over Fuzzy Sets

    Complement

  • Operations over Fuzzy Sets

    Conversely to classic set theory, min (), max (), and 1-id () are not the only possibilities to define logical connectives

    Different functions can be used to represent logical connectives in different situations

    Not only membership functions depend on the context, but also logical connectives!!

  • Fuzzy Complement (c-norms)

    Given a fuzzy set A E, its complement can be defined as follows:

    ( )( ) ,AA C x x E =

    The function C() must satisfy the following conditions:

    (0) 1, (1) 0

    , [0,1], ( ) ( )

    C C

    a b a b C a C b

    = =

  • Fuzzy Complement (c-norms)

    In some cases, two more properties are desirable C(x) is continuous

    C(x) is involutive:

    ( ( )) ,C C a a a E=

    Examples:

    1

    ( ) 1 .

    1( ) (0, )

    1

    ( ) (1 ) (0, )w w

    C x x Std negation

    xC x Sugeno

    x

    C x x w Yager

    =

    =

    =

  • Fuzzy Intersection (t-norms)

    Given two fuzzy sets A, B E, their intersection can be defined as follows:

    [ ]( ) ( ), ( ) ,A B A Bx T x y x y E =

    Required properties:

    ( , ) ( , ) ,

    ( ( , ), ) ( , ( , )) , ,

    ( ), ( ) ( , ) ( , ) , , ,

    ( ,0) 0

    ( ,1)

    T x y T y x x y E commutativity

    T T x y z T x T y z x y z E associativity

    x y w z T x w T y z x y w z E monotony

    T x x E absorption

    T x x x E neutrality

    =

    =

    =

    =

  • Fuzzy Intersection (t-norms)

    Examples:

    ( , ) min( , ) min

    ( , ) max(0, 1)

    ( , )

    min( , ) max( , ) 1( , ) mod

    0

    T x y x y

    T x y x y Lukasiewicz

    T x y x y product

    x y x yT x y product

    otherwise

    =

    = +

    =

    ==

  • Fuzzy Union (t-conorms)

    Given two fuzzy sets A, B E, their union can be defined as follows:

    [ ]( ) ( ), ( ) ,A B A Bx S x y x y E =

    Required properties:

    ( , ) ( , ) ,

    ( ( , ), ) ( , ( , )) , ,

    ( ), ( ) ( , ) ( , ) , , ,

    ( ,1) 1

    ( ,0)

    S x y S y x x y E commutativity

    S S x y z S x S y z x y z E associativity

    x y w z S x w S y z x y w z E monotony

    S x x E absorption

    S x x x E neutrality

    =

    =

    =

    =

  • Fuzzy Union (t-conorms)

    Examples:

    ( , ) max( , ) max

    ( , ) min(1, )

    ( , )

    max( , ) min( , ) 0( , ) mod

    1

    S x y x y

    S x y x y Lukasiewicz

    S x y x y x Y sum

    x y x yS x y sum

    otherwise

    =

    = +

    = +

    ==

  • Properties of Fuzzy Operations

    The t-norms and t-conorms are bounded operators:

    ( , ) min( , ) , [0,1]

    ( , ) max( , ) , [0,1]

    T x y x y x y

    S x y x y x y

    The minimum is the biggest t-norm

    The maximum is the smallest t-conorm

  • Properties of Fuzzy Operations

    Duality (Generalized De Morgan Laws):

    ( ( , )) ( ( ), ( ))

    ( ( , )) ( ( ), ( ))

    C T x y S C x C y

    C S x y T C x C y

    =

    =

    Only some tuples (T, S, C) meet this property

    In such cases the t-norm and the t-conorm are said to be dual w.r.t. the fuzzy complement

    Examples:

    (max, min, 1-id)

    (prod, sum, 1-id)

  • Properties of Fuzzy Operations

    Distributive Properties:

    ( , ( , )) ( ( , ), ( , ))

    ( , ( , )) ( ( , ), ( , ))

    T x S y z S T x y T x z

    S x T y z T S x y S x z

    =

    =

    The only tuple satisfying this property is (max, min, 1-id)

  • Properties of Fuzzy Operations

    In general, given t-norm T, and involutive complement C, we can define operator:

    ( , ) ( ( ( ), ( )))S a b C T C a C b=

    It can be proved that S is a t-conorm s.t. tuple (T, S, C) is dual w.r.t. c-norm C

    Similarly, given S and an involutive C, we can define a dual T for S w.r.t. C as:

    ( , ) ( ( ( ), ( )))T a b C S C a C b=

  • Properties of Fuzzy Operations

    Some dual tuples (T, S, C) satisfy the following properties (excluded-middle and non-contradiction):

    ( , ( ))

    ( , ( ))

    S x C x E

    T x C x

    =

    =

    It can be proved that distributive laws do not hold in such cases

  • Properties of Fuzzy Operations

    Some dual tuples (T, S, C) satisfy the following properties:

    It can be proved that distributive laws do not hold in such cases Except for crisp logic: (max, min, 1-id) are dual (De

    Morgan), distributive, and consistent

    S(x,C(x))=1

    T(x,C(x))=0

    excluded-middle

    non-contradiction

  • Choice of T, S, and C

    The selection of T, S, and C always depend on the concrete case or application

    We need to determine which p