models for inexact reasoning fuzzy logic – lesson 1 ...dia.fi.upm.es/~mgremesal/mir/slides/lesson...
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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
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