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 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 S ystems • Control Units • Bullet train between T okyo 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 { } ⊆ = ∈ = + ∈ | 1 2 , A E n E n k k Z Predicate: “ n is odd”

Crisp Characteristic Functions • The classification of individuals can be done using a indicator or characteristic function: { } µ → : 0,1 E A ∈  1, x A µ =  ( ) x ∉ A  0, x A • Note that: { } − µ = − − 1 (1) K , 3, 1,1,3, K A { } µ − = − − 1 (0) K , 4, 2,0,2,4, K A

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: µ → : [0,1] A E

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 oung, M ature, Old

M embership Functions • M embership functions represent distributions of possibility rather than probability • For instance, the fuzzy set Y oung expresses the possibility that a given individual be young • M embership functions often overlap with each others – 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 { } = , , K , E x x x 1 2 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 n { } = µ = U µ ( ( ) / ) ( ) / A x x x x A A i i = 1 i

Basic Definitions over Fuzzy Sets • Empty set : A fuzzy subset A ⊆ E is empty (denoted A = ø) iff µ = ∀ ∈ ( ) 0, A x x E • Equality : two fuzzy subsets A and B defined over E are equivalent iff µ = µ ∀ ∈ ( ) ( ), x x x E A B

Basic Definitions over Fuzzy Sets • A fuzzy subset A ⊆ E is contained in B ⊆ E iff µ ≤ µ ∀ ∈ ( ) ( ), x x x E A B • Normality: A fuzzy subset A ⊆ E is said to be normal iff µ = max ( ) 1 x A ∈ x E • Support: The support of a fuzzy subset A ⊆ E is a crisp set defined as follows { } = ∈ µ > φ ⊆ ⊆ | ( ) 0 S x E x S E A A A

Operations over Fuzzy Sets • The basic operations over crisp sets can be extended to suit fuzzy sets • Standard operations: – Intersection: µ = µ µ ( ) min( ( ), ( )) x x x ∩ A B A B – Union: µ = µ µ ( ) max( ( ), ( )) x x x ∪ A B A B – Complement: µ = − µ ( ) 1 ( ) A x x A

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 = µ ∀ ∈ C x x E A A • The function C ( ∙ ) must satisfy the following conditions: = = (0) 1, (1) 0 C C ∀ ∈ ≤ → ≥ , [0,1], ( ) ( ) 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 . C x x Std negation − 1 x = λ ∈ ∞ ( ) (0, ) C x Sugeno − λ 1 x 1 = − ∈ ∞ w ( ) (1 ) (0, ) w C x x w Yager

Fuzzy Intersection (t-norms) • Given two fuzzy sets A, B ⊆ E, their intersection can be defined as follows: [ ] µ = µ µ ∀ ∈ ( ) ( ), ( ) , x T x y x y E ∩ A B A B • Required properties: = ∀ ∈ ( , ) ( , ) , 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 = ∀ ∈ ( ,0) 0 T x x E absorption = ∀ ∈ ( ,1) T x x x E neutrality

Fuzzy Intersection (t-norms) • Examples: = ( , ) min( , ) min T x y x y = + − ( , ) max(0, 1) T x y x y Lukasiewicz = ⋅ ( , ) T x y x y product =  min( , ) max( , ) 1 x y x y =  ( , ) mod T x y product  0 otherwise

Fuzzy Union (t-conorms) • Given two fuzzy sets A, B ⊆ E, their union can be defined as follows: [ ] µ = µ µ ∀ ∈ ( ) ( ), ( ) , x S x y x y E ∪ A B A B • Required properties: = ∀ ∈ ( , ) ( , ) , 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 = ∀ ∈ ( ,1) 1 S x x E absorption = ∀ ∈ ( ,0) S x x x E neutrality

Fuzzy Union (t-conorms) • Examples: = ( , ) max( , ) max S x y x y = + ( , ) min(1, ) S x y x y Lukasiewicz = + − ⋅ ( , ) S x y x y x Y sum =  max( , ) min( , ) 0 x y x y =  ( , ) mod S x y sum  1 otherwise

Properties of Fuzzy Operations • The t-norms and t-conorms are bounded operators: ≤ ∀ ∈ ( , ) min( , ) , [0,1] T x y x y x y ≥ ∀ ∈ ( , ) max( , ) , [0,1] 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: 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 M organ), distributive, and “consistent”