# 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

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