Lesson 1 (Fuzzy Sets)

8/3/2019 Lesson 1 (Fuzzy Sets)

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Models for Inexact Reasoning

Fuzzy Logic Lesson 1

Crisp and Fuzzy Sets

Master in Computational Logic

Department of Artificial Intelligence


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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.


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Real-World Applications

ABS Brakes

Expert Systems

Control Units

Bullet train between Tokyo and Osaka

Video Cameras

Automatic Transmissions


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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

E n E n k k Z =

= = +

Z


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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 andrelies on the context in which it is used

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


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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


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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


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Types of Membership Functions

Gaussian


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Triangular


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Trapezoidal


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Example

E = {0, , 100} (Age)

Fuzzy sets: Young, Mature, Old


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Membership Functions

Membership functions represent distributions ofpossibility 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)


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Membership Functions

For practical reasons, in many cases theuniverse of discourse (E) is assumed to be

discrete

{ 1 2, , , n E 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 i

i

A x x x x =

= =U


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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 definedover E are equivalent iff

( ) ( ),A B x x x E =


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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 B x x x =

( ) max( ( ), ( )) A B A B

x x x =

( ) 1 ( )AA

x x =


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Intersection


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Union


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Complement


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Conversely to classic set theory, min (), max(), and 1-id () are not the only possibilitiesto 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!!


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Fuzzy Complement (c-norms)

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

( )( ) ,AA C x x E =

The function C( ) must satisfy the following

conditions:

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

C Ca b a b C a C b

= =


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Fuzzy Complement (c-norms)

In some cases, two more properties aredesirable

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

=

=

=


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Fuzzy Intersection (t-norms)

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

[ ]( ) ( ), ( ) , A B A BT 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

=

=

=

=


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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

=

= + =

==


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Fuzzy Union (t-conorms)

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

[ ]( ) ( ), ( ) , A B A BS 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

=

=

=

=


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Fuzzy Union (t-conorms)

Examples:

( , ) max( , ) max

( , ) min(1, )( , )

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

1

S x y x y

S x y x y LukasiewiczS x y x y x Y sum

x y x yS x y sum

otherwise

=

= += +

==


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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


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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)


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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)


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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=


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

( , ( ))

( , ( ))

S x C x E

T x C x

=

=

It can be proved that distributive laws do nothold in such cases


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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 properties are

required for our application

The most common choice:

T = min, S = max, C = 1-id

Properties:

Comm., assoc., neutrality, absorption, involution, inv. 0-1, inv. 1-0, duality, idempotence, distributive


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Example

Let us suppose that we are thirsty and we arethinking about going to a bar to have a drink

However, we are reluctant to go to whatever

bar We want to go to a bar satisfying the following

requirements:

We want the bar to be traditional

We want to go to a bar close to our home

We want the drinks to be cheap


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Example

To decide to which bar to go, we will make the

following assumptions:

We consider that a bar is traditional if it started

working 5 years or more ago

A bar is close to our home if it is not farther than

ten blocks

A drink is cheap if it costs 1 Euro or less


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Example

We know four different bars to which we can

go:

Price Years Blocks

Bar 1 1.40 3 3

Bar 2 0.80 7 12

Bar 3 1.00 4 9

Bar 4 1.25 5 10


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Example

Using the classical set theory to solve this

problem, we have that the chosen bar must

satisfy the following logical formula:

( ) ( ) ( )5 10 1 years blocks price

This yields the following solution:

Price Years Blocks

Classical

SolutionBar 1 0 0 1 0

Bar 2 1 1 0 0

Bar 3 1 0 1 0

Bar 4 0 1 1 0


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Example

Using the classic set theory we are bounded to

stay at homeL

None of the bars satisfy our requirements!

This is not consistent with the fact we arethirsty

We need a more flexible approach

Let us now try the fuzzy set based approach


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Example

We distinguish three fuzzy sets described by

the following predicates:

The bar is traditional

The bar is close to home The drink is cheap

Thus, first of all we need to model the

abovementioned fuzzy sets

i.e. we need to provide the fuzzy membershipfunctions associated to such fuzzy sets


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Example

MF for the predicate the bar is traditional


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Example

MF for the predicate the bar is close to

home


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Example

Membership function for the predicate the

drink is cheap


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Example

Now, the second step involves the selection ofthe fuzzy operators needed for this application

In this case, we will use the following

operators:

T = min, S = max, C = 1-id

In other cases we will have to carefully choose

the fuzzy operators depending on the required

properties for the concrete application


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Example

Results obtained using fuzzy sets theory:

Price Years Blocks Solution

Bar 1 0,2 0,5 1 0,2

Bar 2 1 1 0,6667 0,6667

Bar 3 1 0,875 1 0,875

Bar 4 0,5 1 1 0,5

Lesson 1 (Fuzzy Sets)

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