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

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 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 Eµ →

Fuzzy Membership Functions

• μA(x) = y

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

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

• μA(x) = 0

– Individual x does not belong to subset A

• μA(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 (μA(x), x), denoted by μA(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 xµ µ=

= = U

Basic Definitions over Fuzzy Sets

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

( ) 0,A x x Eµ = ∀ ∈

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

( ) ( ),A Bx x x Eµ µ= ∀ ∈

Basic Definitions over Fuzzy Sets

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

( ) ( ),A Bx x x Eµ µ≤ ∀ ∈

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

max ( ) 1Ax E

xµ∈

=

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

{ }| ( ) 0A A AS x E x S Eµ 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 Bx x xµ µ µ∩ =

( ) max( ( ), ( ))A B A Bx x xµ µ µ∪ =

( ) 1 ( )AAx xµ 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:

( )( ) ,AAC x x Eµ µ= ∀ ∈

• 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

λλ

= −

−= ∈ ∞

−

= − ∈ ∞

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 Eµ µ µ∩ = ∀ ∈

• 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 Eµ 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( , ) 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 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:

Price Years Blocks

Bar 1 1.40 3 3

Bar 2 0.80 7 12

Bar 3 1.00 4 9Bar 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 1years blocks price≥ ∧ ≤ ∧ ≤

• This yields the following solution:

Price Years Blocks

Classical

Solution

Bar 1 0 0 1 0

Bar 2 1 1 0 0

Bar 3 1 0 1 0Bar 4 0 1 1 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:

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