M odels for Inexact Reasoning Fuzzy Logic Lesson 1 Crisp and Fuzzy - - PowerPoint PPT Presentation

M odels for Inexact Reasoning Fuzzy Logic – Lesson 1 Crisp and Fuzzy Sets

M aster 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
• f 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 S

ystems

• Control Units
• Bullet train between T
• kyo 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 A x x A µ µ → ∈  =  ∉ 

{ } { }

1 1

(1) , 3, 1,1,3, (0) , 4, 2,0,2,4,

A A

µ µ

− −

= − − = − − 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 M embership 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

µ →

Fuzzy M embership 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 M embership Functions

• Gaussian

Types of M embership Functions

• Triangular

Types of M embership Functions

• Trapezoidal

Example

• E = {0, …

, 100} (Age)

• Fuzzy sets: Y
• ung, M ature, Old

M embership Functions

• M embership functions represent distributions of

possibility rather than probability

• For instance, the fuzzy set Y
• ung expresses the

possibility that a given individual be young

• M embership functions often overlap with each
• thers

– A given individual may belong to different fuzzy sets

(with different degrees)

M embership Functions

• For practical reasons, in many cases the

universe of discourse (E) is assumed to be discrete

{ }

1 2

, , ,

n

E x x x = K

• 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 { }

1

( ( ) / ) ( ) /

n A A i i i

A x x x x µ µ

=

= =U

Basic Definitions over Fuzzy Sets

• Empty set: A fuzzy subset A ⊆ E is empty

(denoted A = ø) iff

( ) 0,

A x

x E µ = ∀ ∈

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

( ) ( ),

A B

x x x E µ µ = ∀ ∈

Basic Definitions over Fuzzy Sets

• A fuzzy subset A ⊆ E is contained in B ⊆ E iff

( ) ( ),

A B

x x x E µ µ ≤ ∀ ∈

• Normality: A fuzzy subset A ⊆ E is said to be

normal iff

max ( ) 1

A x E

x µ

∈

=

• Support: The support of a fuzzy subset A ⊆ E

is a crisp set defined as follows

{ }

| ( )

A A A

S x E x S E µ φ = ∈ > ⊆ ⊆

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 µ µ µ

∩

= ( ) max( ( ), ( ))

A B A B

x x x µ µ µ

∪

=

( ) 1 ( )

A A x

x µ µ = −

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:

( )

( ),

A A

C x x E µ µ = ∀ ∈

• The function C(∙ ) must satisfy the following

conditions:

(0) 1, (1) , [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 x C x Sugeno x C x x w Yager λ λ = − − = ∈ ∞ − = − ∈ ∞

Fuzzy Intersection (t-norms)

• Given two fuzzy sets A, B ⊆ E, their

intersection can be defined as follows:

[ ]

( ) ( ), ( ) ,

A B A B

x T x y x y E µ µ µ

∩

= ∀ ∈

• Required properties:

( , ) ( , ) , ( ( , ), ) ( , ( , )) , , ( ),( ) ( , ) ( , ) , , , ( ,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 T x y x y T x y x y Lukasiewicz T x y x y product x y x y T x y product

• therwise

= = + − = ⋅ =  =  

Fuzzy Union (t-conorms)

• Given two fuzzy sets A, B ⊆ E, their union can

be defined as follows:

[ ]

( ) ( ), ( ) ,

A B A B

x S x y x y E µ µ µ

∪

= ∀ ∈

• 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( , ) ( , ) mod 1 S x y x y S x y x y Lukasiewicz S x y x y x Y sum x y x y S x y sum

• therwise

= = + = + − ⋅ =  =  

Properties of Fuzzy Operations

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

( , ) 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 M organ 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

M organ), 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 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

• T
• 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:

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

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 Solution Bar 1 1 Bar 2 1 1 Bar 3 1 1 Bar 4 1 1

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

• M F for the predicate “ the bar is traditional”

Example

• M F for the predicate “ the bar is close to

home”

Example

• M embership 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
• perators:

– 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:

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