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Chap 4: Fuzzy Expert SystemsPart 1

Asst. Prof. Dr. Sukanya PongsuparbDr. Srisupa Palakvangsa Na AyudhyaDr. Benjarath Pupacdi

SCCS451 Artificial IntelligenceWeek 9

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Agenda

Traditional LogicWhat is Fuzzy Logic?Boolean Logic VS Fuzzy LogicWhat is Fuzzy Set?Linguistic Variable and HedgesOperations of Fuzzy SetsProperties of Fuzzy Sets

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Logic

Is it hot or cold?Do you like playing crossword?Do you like Harry Potter or not?

ONE ZERO

Only two values

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Represented Using Set

HOT

COLD

Do notLike

Harry

Like Harry

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Digital gates

Only two inputs: zero and oneOnly two outputs: zero and one

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What do you think of the AI midterm exam?

Definitely easyReally easyVery very easyVery easyEasyQuite easy

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Doctors say…..

You could possibly have a cold.You are certainly have chicken pox.It is likely that you may have ไข้�หวั�ด 2009.

Someone says…..It is quite likely that I cannot go to a partyIt looks like it is going to rainI am certain that Mum will not like this bag

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Which one is Monkey?

Which one is Human?

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“Fuzzy” = Unclear

veryreally quite

more or lessvery very extremely

VAGUE & AMBIGUOUS

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Fuzzy Logic

Unclear

The formal systematic study of the principles of valid inference and correct reasoning

a form of multi-valued logic derived from fuzzy set theory to deal with reasoning that is approximate rather than precise. (Wikipedia)

a theoretical system used in mathematics, computing and philosophy to deal with statements which are neither true nor false (dictionary.cambridge.org)

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“Fuzzy logic” is not logic that is fuzzybut logic that is used to

describe fuzziness

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“Fuzzy logic” is determined as a set of mathematical principles for knowledge

representation based on degrees of membership rather than on

crisp membership of classical binary logic

Zadeh 1965(Master of Fuzzy Logic)

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Boolean Logic VS Fuzzy Logic

Boolean Logic Fuzzy LogicTwo – Valued Logic: Zero & One

Multi – Valued Logic

Sharp boundary(crisp)

Range of valuesDegree of memberships degree of truthModel senses of words

(a) Boolean Logic. (b) Multi-valued Logic.0 1 10 0.2 0.4 0.6 0.8 100 1 10

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Degree of Membership

Fuzzy

Mark

John

Tom

Bob

Bill

1

1

1

0

0

1.00

1.00

0.98

0.82

0.78

Peter

Steven

Mike

David

Chris

Crisp

1

0

0

0

0

0.24

0.15

0.06

0.01

0.00

Name Height, cm

205

198

181

167

155

152

158

172

179

208

Who is tall?

Tall

Short

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150 210170 180 190 200160

Height, cmDegree ofMembership

Tall Men

150 210180 190 200

1.0

0.0

0.2

0.4

0.6

0.8

160

Degree ofMembership

170

1.0

0.0

0.2

0.4

0.6

0.8

Height, cm

Fuzzy Sets

Crisp Sets

Let’s the graph.....

X = Universe of Discourse

Y = Membership Value

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Classic Paradoxes of Logic

Q: Does the Cretan philosopher tell the truth when he asserts that “All Cretans always lie’?

Boolean Logic: This assertion contains a contradiction.Fuzzy Logic: The philosopher does and does not tell the

truth!

The barber of a village gives a hair cut only to those whodo not cut their hair themselves?Q: Who cuts the barber’s hair?Boolean Logic: This assertion contains a contradiction.Fuzzy Logic: The barber cuts and does not cut his own hair!

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Set Representation

Use a concept of Set to represent the idea of classical set and fuzzy set

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Classical (Crisp) Set

fA (x) called the characteristic function of APrinciple of Dichotomy: a classic set theory imposes a sharp boundary

fA(x): X {0, 1}, where

Ax

Axxf A if0,

if 1,)(

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Fuzzy Set

A(x): X [0, 1]where

A(x) = 1 if x is totally in A;A(x) = 0 if x is not in A;0 < A(x) < 1 if x is partly in A.

A(x) : membership function of set Amembership value (0<= degree <=1): shows the degree of membershipBasic idea: an element belongs to a fuzzy set with a certain degree of membershipfuzzy set is a set with fuzzy boundaries

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How to represent a fuzzy set in a computer?

1. Define the membership function2. Perform knowledge acquisition from…

1) Single expert2) Multiple experts

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Example of Tall Man

1. Define the membership function: short, average, tall2. Knowledge Acquisition

1) Fuzzy Sets: short, average, tall2) Universe of Discourse (height): short, average, tall

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Example of Tall Man (cont.)

150 210170 180 190 200160

Height, cmDegree ofMembership

Tall Men

150 210180 190 200

1.0

0.0

0.2

0.4

0.6

0.8

160

Degree ofMembership

Short Average ShortTall

170

1.0

0.0

0.2

0.4

0.6

0.8

Fuzzy Sets

Crisp Sets

Short Average

Tall

Tall

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Example of Tall Man (cont.)

Fuzzy Subset A

Fuzziness

1

0Crisp Subset A Fuzziness x

X

(x)

Representation of crisp and fuzzy subset of X

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Example of Tall Man (cont.)

Crisp SetLet X be the universe of discourse

X = {x1, x2, x3, x4, x5}

Let A be a crisp subset of X, A = {x2, x5}A can be described using a set of pair {(xi, A(xi)} where A(xi) is the membership function

A = { (x1,0), (x2,1), (x3,0), (x4,0), (x5,0) }

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Example of Tall Man (cont.)

Fuzzy SetA can be a fuzzy subset of X if and only if,

A = { (x, A(x) } x Є X, A(x): X [0, 1]

it can be re-written as

A = {A(x1) / x1}, {A(x2) / x2}, ….., {A(xn) / xn}e.g.

Tall man = (0/180, 0.5/185, 1/190)Short man = (1/160, 0.5/165, 0/170)

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Example of Tall Man (cont.)

Fuzzy sets must be represented as functions and then mapped the elements of the sets to their degree of membershipExamples of functions:-

SigmoidGaussianPi

In practice, most applications use linear fit functions (shown in Slide No. 20)

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Typical Membership Functions

Sigmoid function

Gaussian function

Pi function

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Membership Functions: Linear fit functions

Trapezoidal function

Triangular function

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Membership Functions: Trapezoidal function

Triangular function

The trapezoidal curve is a function of a vector x, and depends on four scalar parameters a, b, c, and d, as given by

Trapezoidal function

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Membership Functions: Triangular function

The triangular curve is a function of a vector x, and depends on three scalar parameters a, b, and c, as given by

Triangular function

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How to convert “words” to the degree of membership?

Linguistic Variables & Hedges

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Linguistic Variables

IF <antecedent> THEN <consequent>

valueobject valueobject

Examples:• IF wind is strong THEN sailing is good• IF speed is slow THEN stopping_distance is short• IF project_duration is long THEN completion_risk is high

Linguistic variable Linguistic value

* linguistic variable == fuzzy variable

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Hedges

Andy quite likes Thai foodhedges

Mary looks very much like her mother

Jim has been to several attractions in Thailand

Hedges: terms that modify the shape of fuzzy sets e.g. very, somewhat, quite, more or less, and slightly

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Hedges (cont.)

Hedges can modify verbs, adjectives, adverbs, or whole sentences. They are used as

All-purpose modifiers: very, quite, extremelyTruth-values: quite true, mostly falseProbabilities: likely, not very likelyQuantifiers: most, several, fewPossibilities: almost impossible, quite possible

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Example: Fuzzy sets with the hedge “very”

Short

Very Tall

Short Tall

Degree ofMembership

150 210180 190 200

1.0

0.0

0.2

0.4

0.6

0.8

160 170

Height, cm

Average

TallVery Short Very Tall

A man who is 185 cm tall is a member of the tall men set with a degree of membership of 0.5. He is also a member of the very tall men set with a degree of 0.15.

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Hedges (cont.)

There are two types of “hedges”:-Concentration

reduce the size of the fuzzy set e.g. very, very very, extremely, slightlydecrease the degree of membership

Dilationexpand the size of the fuzzy set e.g. more or less, somewhatIncrease the degree of the membership

IntensificationIntensifies the meaning of the whole sentence e.g. indeedIncrease the degree of the membership above 0.5 and decrease those below 0.5

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Representation of hedges in fuzzy logicHedge Mathematical

Expression

A little

Slightly

Very

Extremely

Hedge MathematicalExpression Graphical Representation

[A ( x )]1.3

[A ( x )]1.7

[A ( x )]2

[A ( x )]3

Concentration

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Representation of hedges in fuzzy logic (cont.)Hedge Mathematical

ExpressionHedge MathematicalExpression Graphical Representation

Very very

More or less

Indeed

Somewhat

2 [A ( x )]2

A ( x )

A ( x )

if 0 A 0.5

if 0.5 < A 1

1 2 [1 A ( x )]2

[A ( x )]4 Concentration

dilation

dilation

intensification

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What can we do withthese formulas (?_?)

To expand or reduce the subset by modifying the degree of membership

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Examples

If Alice has a 0.86 membership in the set of tall girls

the equation of “very” is..

A(x) of very = [A(x)]2

So Alice will have a [0.86]2 = 0.7396 membership in the set of very tall girls

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Examples (cont.)

If Alice has a 0.86 membership in the set of tall girls

the equation of “very very” is..

A(x) of very very = [A(x) of very] 4 = [A(x)]4

So Alice will have a [0.86]4 = 0.5470 membership in the set of very very tall girls

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Operations of fuzzy sets

The classical set theory developed in the late 19th century by Georg Cantor describes how crisp sets caninteract. These interactions are called operations:

- Complement- Containment- Intersection- Union

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Complement

Complement is the opposite set of a given set e.g. if a set is X, the complement set is NOT X

Questions:-

Crisp Sets: Who does not belong to the set?

Fuzzy Sets: How much do elements not belong to the set?

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Complement (cont.)

EquationA(x) = 1 A(x)

Exampletall man = (0/180, 0.25/182.5, 0.5/185, 0.75/187.5, 1/190)

NOTtall man = (1/180, 0.75/182.5, 0.5/185, 0.25/187.5, 0/190)

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Containment

Containment: one set is the subset of another set

Questions:-

Crisp Sets: Which sets belong to which other sets?

Fuzzy Sets: Which sets belong to other sets?

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Containment (cont.)

In crisp set, all elements of a subset entirely belong to a larger set i.e. the degree of membership is equal to 1.In fuzzy sets, each element can belong less to the subset than to the larger set.Example

tall man = (0/180, 0.25/182.5, 0.5/185, 0.75/187.5, 1/190)

verytall man = (0/180, 0.06/182.5, 0.25/185, 0.56/187.5, 1/190)

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Intersection

Intersection: the area where sets overlapIn crisp sets, an element must belong to both sets In fuzzy sets, an element may partly belong to both sets with different memberships

Questions:-Crisp Sets: Which element belongs to both sets?

Fuzzy Sets: How much of the element is in both sets?

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Intersection (cont.)

EquationAB(x) = min [A(x), B(x)] = A(x) B(x), where xX

tall man = (0/165, 0/175, 0.0/180, 0.25/182.5, 0.5/185, 1/190)averagetall man = (0/165, 1/175, 0.5/180, 0.25/182.5, 0.0/185, 0/190)

tall man average man = (0/165, 0/175, 0.0/180, 0.25/182.5, 0.0/185, 0/190)

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Intersection (cont.)

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Union

Union: the integration area of all setsIn crisp sets, an element can belong to either sets.In fuzzy sets, the union is the largest membership value of the element in either set.

Questions:-Crisp Sets: Which element belongs to either set?

Fuzzy Sets: How much of the element is in either set?

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Union (cont.)

EquationAB(x) = max [A(x), B(x)] = A(x) B(x), where xX

tall man = (0/165, 0/175, 0.0/180, 0.25/182.5, 0.5/185, 1/190)averagetall man = (0/165, 1/175, 0.5/180, 0.25/182.5, 0.0/185, 0/190)

tall man average man = (0/165, 1/175, 0.5/180, 0.25/182.5, 0.5/185, 1/190)

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Operations of fuzzy sets

Complement

0x

1

( x )

0x

1

Containment

0x

1

0x

1

A B

Not A

A

Intersection

0x

1

0x

A B

Union0

1

A BA B

0x

1

0x

1

B

A

B

A

( x )

( x )

( x )

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Set Properties

The properties of set used in crisp sets can also be used in fuzzy setsFrequently used properties:-

CommutativityAssociativelyDistributivityIdempotencyIdentityInvolutionTransitivityDe Morgan’s Laws

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Set Properties (cont.)

CommutativityA B = B A

AssociativelyA (B C) = (A B ) CA (B C) = (A B ) C

DistributivityA (B C) = (A B ) (A C)A (B C) = (A B ) (A C)

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Set Properties (cont.)

IdempotencyA A = AA A = A

Identity A Ф = AA X = AA Ф = ФA X = X

where Ф is an empty set and X is a superset of A.

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Set Properties (cont.)

Involution ( A) = A

Transitivity If (A B) ⊂ (B C) then A C⊂ ⊂

De Morgan’s LawsØ (A B) = A BØ (A B) = A B

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How can we use theseconfusing stuff (?_?)

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Example

Properties and hedges can be used to obtain a variety of fuzzy sets from the existing ones.Assume that we have a fuzzy set A of tall men, we can derive a fuzzy set of very tall man:-

Given fuzzy set A of tall men = A(x)

very tall men = [A(x)]2

NOT very tall men A(x) = 1 - [A(x)]2

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Exercise

Givenfuzzy set A of raining day = A(x)fuzzy set B of cold day = B(x)

Derive a fuzzy set C of extremely cold and not raining dayDerive a fuzzy set D of not very very cold or slightly raining day