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

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:

E

= Z

A

{ ∈

Z

}

Crisp Characteristic Functions

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

µ

A

µ

A

: E

= 

1,

0, x

A x

A

• Note that:

µ

µ

A

1

(1)

A

1

(0)

= {

K

, 3, 1,1, 3,

K

}

= {

K

, 4, 2, 0, 2, 4,

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:

µ

A

: E

[0,1]

Fuzzy Membership Functions

• μ

A

(x) = y

– Individual x belongs to some extent (“y”) to subset

A

– y is the degree to which the individual x is tall

• μ

A

(x) = 0

– Individual x does not belong to subset A

• μ

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

E

= {

,

1 2

,

K

, x n

}

• The pair (μ

A

(x), x), denoted by μ

A

(x)/x is called fuzzy singleton

• Fuzzy sets can be described in terms of fuzzy singletons

A

= {

(

µ

A x x

} = i n

U

=

1

µ

A

( ) / x i

Basic Definitions over Fuzzy Sets

• Empty set : A fuzzy subset A

E is empty

(denoted A = ø) iff

µ

A

( )

=

0,

∀ ∈

E

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

µ

A x

=

µ

B x x E

Basic Definitions over Fuzzy Sets

• A fuzzy subset A

E is contained in B

E iff

µ

A

≤ µ

B x E

• Normality: A fuzzy subset A

E is said to be normal iff max

µ

A x

=

1

• Support: The support of a fuzzy subset A

E is a crisp set defined as follows

S

A

{

E |

µ

A x

>

0

} φ ⊆

S

A

E

Operations over Fuzzy Sets

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

• Standard operations:

– Intersection:

µ x

= min(

µ

A x

µ

B x

– Union:

µ x

= max(

µ

A x

µ

B x

– Complement:

µ

A x

= − µ

A 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:

µ

A

=

C

(

µ

A

)

, x E

• The function C (∙) must satisfy the following conditions:

C (0)

=

1, C (1)

=

0

,

[0,1], a

≤ →

( )

( )

Fuzzy Complement (c-norms)

• In some cases, two more properties are desirable

– C(x) is continuous

– C(x) is involutive:

( ( ))

= a ,

∀ ∈

E

• Examples: x

=

1

1

− λ x x

C x

= − x w

)

1 w

λ ∈

(0, ) w

(0, )

Sugeno

Yager

Fuzzy Intersection (t-norms)

• Given two fuzzy sets A, B

E, their intersection can be defined as follows:

µ

( )

=

T

[ µ

A x

µ

B y

] ∀

,

E

• Required properties:

( , )

=

( , )

,

E commutativity

( ( , ), )

=

( , ( , ))

, ,

E associativity

( x

≤ y ), ( w z ) ( , )

( , )

, , ,

E monotony

( , 0)

=

0 x E absorption

( ,1)

= x

∀ ∈

E neutrality

Fuzzy Intersection (t-norms)

• Examples:

( , )

= x y

( , )

= max(0, x y 1)

( , )

= ⋅ min( , ) x y

=

0 otherwise min

Lukasiewicz product mod product

Fuzzy Union (t-conorms)

• Given two fuzzy sets A, B

E, their union can be defined as follows:

µ

( )

=

S

[ µ

A

( ),

µ

B y

] ∀

,

E

• Required properties:

( , )

=

( , )

,

E commutativity

( ( , ), )

=

( , ( , ))

, ,

E associativity

( x

≤ y ), ( w z ) ( , )

( , )

, , ,

E monotony

= x E absorption

= x x E neutrality

Fuzzy Union (t-conorms)

• Examples:

( , )

= x y

( , )

= + − ⋅ max

( , )

= min(1, x

+ y ) Lukasiewicz sum max( , ) x y

=

0 mod sum

1 otherwise

Properties of Fuzzy Operations

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

( , )

≤ x y

( , )

≥ x y

,

[0,1]

,

[0,1]

• The minimum is the biggest t-norm

• The maximum is the smallest t-conorm

Properties of Fuzzy Operations

• Duality (Generalized De Morgan Laws):

( ( , ))

=

( ( ), ( ))

( ( , ))

=

( ( ), ( ))

• 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:

( , ( , ))

=

( ( , ), ( , ))

( , ( , ))

=

( ( , ), ( , ))

• 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:

( , )

=

( ( ( ), ( )))

• 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:

( , )

=

( ( ( ), ( )))

Properties of Fuzzy Operations

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

( , ( ))

=

E

= ∅

• 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:

S(x,C(x ))=1 excluded-middle

T(x,C(x ))=0 non-contradiction

• 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”

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

Example

• Let us suppose that we are thirsty and we are thinking 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

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

Example

• We know four different bars to which we can go:

Bar 1

Bar 2

Bar 3

Bar 4

Price

1.40

0.80

1.00

1.25

Years

3

7

4

5

Blocks

3

12

9

10

Example

• Using the classical set theory to solve this problem, we have that the chosen bar must satisfy the following logical formula:

( years

) ( blocks

10

) ( price

1

)

• This yields the following solution:

Bar 1

Bar 2

Bar 3

Bar 4

Price

0

1

1

0

Years

0

1

0

1

Blocks

1

0

1

1

Classical

Solution

0

0

0

0

Example

• Using the classic set theory we are bounded to stay at home L

– None of the bars satisfy our requirements!

• This is not consistent with the fact “we are thirsty”

• We need a more flexible approach

• Let us now try the fuzzy set based approach

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 membership functions associated to such fuzzy sets

Example

• MF for the predicate “the bar is traditional”

Example

• MF for the predicate “the bar is close to home”

Example

• Membership function for the predicate “the drink is cheap”

Example

• Now, the second step involves the selection of the 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

Example

• Results obtained using fuzzy sets theory:

Bar 1

Bar 2

Bar 3

Bar 4

Price

0,2

1

1

0,5

Years

0,5

1

0,875

1

Blocks

1

0,6667

1

1

Solution

0,2

0,6667

0,875

0,5

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