fuzzy logic

FUZZY LOGIC

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 sets

• In the 70s: From FST to Fuzzy Logic

• Nowadays: Applications to control systems

– Industrial applications

– Domotic applications, etc.

What is a set

Classical Sets

Classical sets – either an element belongs to the set or it does not.

For example, for the set of integers, either an integer is even or it is not (it is odd).

Classical sets are also called crisp (sets).

Lists: A = {apples, oranges, cherries, mangoes}

A = {a1,a2,a3 }

A = {2, 4, 6, 8, …}

Formulas: A = {x | x is an even natural number}

A = {x | x = 2n, n is a natural number}

Membership or characteristic function

Ax

Axx

A if 0

if 1)(

Classical Sets

Introduction to fuzzy logic

Introduction to fuzzy logic

Fuzzy Sets

Sets with fuzzy boundaries

A = Set of tall people

Heights5’10’’

1.0

Crisp set A

Membershipfunction

Heights5’10’’6’2’’

.5

.9

Fuzzy set A1.0

Membership Functions (MFs) Characteristics of MFs:

Subjective measures Not probability functions

MFs

Heights5’10’’

.5

.8

.1

“tall” in Asia

“tall” in the US

“tall” in NBA

Fuzzy Sets

Formal definition:A fuzzy set A in X is expressed as a set of

ordered pairs:

A x x x XA {( , ( ))| }

Universe oruniverse of discourse

Fuzzy setMembership

function(MF)

A fuzzy set is totally characterized by amembership function (MF).

An Example• A class of students

(E.G. MCA. Students taking „Fuzzy Theory”)• The universe of discourse: X

• “Who does have a driver’s licence?”• A subset of X = A (Crisp) Set• (X) = CHARACTERISTIC FUNCTION

• “Who can drive very well?” (X) = MEMBERSHIP FUNCTION

1 0 1 1 0 1 1

0.7 0 1.0 0.8 0 0.4 0.2

Crisp or Fuzzy Logic Crisp Logic

A proposition can be true or false only.• Bob is a student (true)• Smoking is healthy (false)

The degree of truth is 0 or 1.

Fuzzy Logic The degree of truth is between 0 and 1.

• William is young (0.3 truth) • Ariel is smart (0.9 truth)

Fuzzy Sets Crisp Sets

Membership Characteristic

function function

]1,0[XA }1,0{XmA

Set-Theoretic Operations

Set-Theoretic Operations Subset

Complement

Union

Intersection

( ) ( ), A BA B x x x U

( ) max( ( ), ( )) ( ) ( )C A B A BC A B x x x x x

( ) min( ( ), ( )) ( ) ( )C A B A BC A B x x x x x

( ) 1 ( )AAA U A x x

Set-Theoretic Operations

A B

A B

A

A B

Properties Of Crisp Set

A AInvolution

A B B A Commutativity

A B B A

A B C A B C Associativity A B C A B C

A B C A B A C Distributivity A B C A B A C

A A A Idempotence

A A A

A A B A Absorption A A B A

A B A B

De Morgan’s laws

A B A B

Properties The following properties are invalid

for fuzzy sets: The laws of contradiction

The laws of exclude middleA A

A A U

Properties of Fuzzy Set

height0

1

short tall

Example we have two discrete fuzzy sets

Example (cont..)

Summarize propertiesInvolution

Commutativity AB=BA, AB=BA

AssociativityABC=(AB)C=A(BC),ABC=(AB)C=A(BC)

DistributivityA(BC)=(AB)(AC),A(BC)=(AB)(AC)

Idempotence AA=A, AA=A

Absorption A(AB)=A, A(AB)=A

Absorption of complement

Abs. by X and AX=X, A=Identity A=A, AX=A

Law of contradiction

Law of excl. middle

DeMorgan’s laws

AA

BABAA

BABAA

AA

XAA

BABABABA

Fuzzy and Crisp

Operations

Fuzzy Set Operations

Crisp Set Operations

Representation of Crisp set

Fuzzy and crisp Relations

CARTESIAN PRODUCT

An ordered sequence of r elements, written in the form (a1, a2, a3, . . . , ar), is called an ordered r-tuple.

For crisp sets A1,A2, . . . ,Ar, the set of all r-tuples(a1, a2, a3, . . . , ar), where a1 A1,a2 A2, ∈ ∈and ar Ar, is called the ∈ Cartesian product of A1,A2, . . . ,Ar, and is denoted by

A1×A2×···×Ar.

(The Cartesian product of two or more sets is not the same thing as the arithmetic product of two or more sets.)

Crisp Relations

A subset of the Cartesian product A1×A2×···×Ar is called an r-ary relation over A1,A2, . . . ,Ar.

If three, four, or five sets are involved in a subset of the full Cartesian product, the relations are called ternary, quaternary, and quinary

Cartesian product

The Cartesian product of two universes X and Y is determined as

X ×Y = {(x, y) | x X, y Y}∈ ∈ which forms an ordered pair of every x X with ∈

every y Y, forming unconstrained∈ matches between X and Y. That is, every element

in universe X is related completely to every element in universe Y.

Fuzzy Relations

Triples showing connection between two sets:

(a,b,#): a is related to b with degree #

Fuzzy relations are set themselves

Fuzzy relations can be expressed as matrices

…

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Fuzzy Relations Matrices

Example: Color-Ripeness relation for tomatoes

R1(x, y) unripe semi ripe ripe

green 1 0.5 0

yellow 0.3 1 0.4

Red 0 0.2 1

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Fuzzy Relations

A fuzzy relation R is a 2D MF:

( ,,( , ) ( , )) |R x yx y x yR X Y

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Fuzzy relation

A fuzzy relation is a fuzzy set defined on the Cartesian product of crisp sets A1, A2, ..., An where tuples (x1, x2, ..., xn) may have varying degrees of membership within the relation.

The membership grade indicates the strength of the relation present between the elements of the tuple.

1 2

1 2 1 2 1 1 2 2

: ... [0,1]

(( , ,..., ), ) | ( , ,..., ) 0, , ,..., R n

n R R n n n

A A A

R x x x x x x x A x A x A

Compositions

A fuzzy relation defined on X an Z.

Max-Min Composition

X Y ZR: fuzzy relation defined on X and Y.

S: fuzzy relation defined on Y and Z.

R。 S: the composition of R and S.

( , ) max min ( , ), ( , )R S y R Sx z x y y z

( , ) ( , )y R Sx y y z

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Max-min composition

Example

( , ) max[min( ( , ), ( , ))]

[ ( , ) ( , )]

S R R Sy

R Sy

x z x y y z

x y y z

( , ) , ( , )x y A B y z B C

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Example

(1, ) max[min(0.1,0.9),min(0.2,0.2),min(0.0,0.8),min(1.0,0.4)]

max[0.1,0.2,0.0,0.4] 0.4S R

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Example

(1, ) max[min(0.1,0.0),min(0.2,1.0),min(0.0,0.0),min(1.0,0.2)]

max[0.0,0.2,0.0,0.2] 0.2S R

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Max-Product Composition

( , ) max ( , ) ( , )R S v R Sx y x v v y

A fuzzy relation defined on X an Z.

X Y ZR: fuzzy relation defined on X and Y.

S: fuzzy relation defined on Y and Z.

R。 S: the composition of R and S.

Max-min composition is not mathematically tractable, therefore other compositions such as max-product composition have been suggested.

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Max Product

Max product: C = A・ B=AB= Example

12 ?C

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Max product

Example

12 0.1C

45

Max product

Example

13 0.5C

46

Max product

Example

C

Fuzzy System

Introduction

Fuzzify crisp inputs to get the fuzzy inputs Defuzzify the fuzzy outputs to get crisp outputs

Fuzzy Systems

Fuzzy Knowledge base

Input FuzzifierInferenceEngine

Defuzzifier Output

Propositional logic

A proposition is a statement- in which English is a declarative sentence and logic defines the way of putting symbols together to form a sentences that represent facts

Every proposition is either true or false.

Example of PL

The conjunction of the two sentences: Grass is greenPigs don't flyis the sentence: Grass is green and pigs don't flyThe conjunction of two sentences will

be true if, and only if, each of the two sentences from which it was formed is true.

Statement symbols and variables

Statement: A simple statement is one

that does not contain any other statement as a part.

A compound statement is one that has two or more simple statement as parts called components.

Symbols for connective

ASSERTION P “P IS TRUE”

NEGATION ¬P ~

! NOT “P IS FALSE”

CONJUCTION

P^Q . & && AND “BOTH P AND Q ARE TRUE

DISJUNCTION

PvQ || | OR “ EITHER P OR Q IS TRUE”

IMPLICATION P-> Q ⇒ IF…THEN

“IF P IS TRUE THEN Q IS TRUE.”

EQUIVALENCE

P⇔Q =⇔

IF AND ONLY IF

“P AND Q ARE EITHER BOTH TRUE OR FALSE”

Truth Value

The truth value of a statement is truth or falsity.

P is either true or false ~p is either true of false p^q is either true or false, and

so on. Truth table is a convenient way of

showing relationship between several propositions..

Case 1

Case 2

Truth Table for Negation

As you can see “P” is a true statement then its negation “~P” or “not P” is false.

If “P” is false, then “~P” is true.

P

T

TF

F

~P

Truth Table for Conjunction

P QCase 1

Case 2

Case 3

Case 4

T

T

F

F

T

F

T

F

T

F

F

F

P Λ Q

Truth Table for Disjunction

P QCase 1

Case 2

Case 3

Case 4

T

T

F

F

T

F

T

F

T

T

T

F

P V Q

Tautology

Tautology is a proposition formed by combining other proposition (p,q,r…)which is true regardless of truth or falsehood of p,q,r…

DEF: A compound proposition is called a tautology if no matter what truth values its atomic propositions have, its own truth value is T.

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Tautology example

Demonstrate that

[¬p (p q )]qis a tautology in two ways:

1. Using a truth table – show that [¬p (p q )]q is always true.

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Tautology by truth table

p q ¬p p q ¬p (p q ) [¬p (p q )]q

T T

T F

F T

F F

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Tautology by truth table

p q ¬p p q ¬p (p q ) [¬p (p q )]q

T T F

T F F

F T T

F F T

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Tautology by truth table

p q ¬p p q ¬p (p q ) [¬p (p q )]q

T T F T

T F F T

F T T T

F F T F

63L3

Tautology by truth table

p q ¬p p q ¬p (p q ) [¬p (p q )]q

T T F T F

T F F T F

F T T T T

F F T F F

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Tautology by truth table

p q ¬p p q ¬p (p q ) [¬p (p q )]q

T T F T F T

T F F T F T

F T T T T T

F F T F F T

Modus Ponens and Modus Tollens

Modus ponens -- If A then B, observe A, conclude B

Modus tollens – If A then B, observe not-B, conclude not-A

Modus Ponens and Tollens

If Joan understood this book, then she would get a good grade. If P then Q Joan understood .: she got a good grade. This uses modus ponens. P .: Q

If Joan understood this book, then she would get a good grade. If P then Q She did not get a good grade .: she did

not understand this book. ~Q .: ~P This uses modus tollens.

Fuzzy QuantifiersThe scope of fuzzy propositions can be

extended using fuzzy quantifiers

• Fuzzy quantifiers are fuzzy numbers that take

part in fuzzy propositions

• There are two different types:

– Type #1 (absolute): Defined on the set of real

numbers

• Examples: “about 10”, “much more than 100”, “at least

about 5”, etc.

– Type #2 (relative): Defined on the interval [0, 1]

• “almost all”, “about half”, “most”, etc.

Fuzzification

The fuzzification comprises the process of transforming crisp values into grades of membership for linguistic terms of fuzzy sets. The membership function is used to associate a grade to each linguistic term.

Measurement devices in technical systems provide crisp measurements, like 110.5 Volt or 31,5 °C. At first, these crisp values must be transformed into linguistic terms (fuzzy sets) . This is called fuzzification.

FuzzifierFuzzy

Knowledge baseFuzzy

Knowledge base

I nput FuzzifierI nference

EngineDefuzzifier OutputI nput Fuzzifier

I nferenceEngine

Defuzzifier Output

Converts the crisp input to a linguistic

variable using the membership functions

stored in the fuzzy knowledge base.

Fuzzy interference

If x is A and y is B then z = f(x, y)

Fuzzy Sets Crisp Function

f(x, y) is very often a

polynomial function w.r.t. x

and y.

Examples

R1: if X is small and Y is small then z = x +y +1

R2: if X is small and Y is large then z = y +3

R3: if X is large and Y is small then z = x +3

R4: if X is large and Y is large then z = x + y + 2

Defuzzification

• Convert fuzzy grade to Crisp output The max criterion method finds the point at

which the membership function is a maximum.

The mean of maximum takes the mean of those points where the membership function is at a maximum.

Defuzzification