CS621: Introduction to Artificial Intelligence - PowerPoint Presentation

Slide1

CS621: Introduction to Artificial Intelligence

Pushpak Bhattacharyya

CSE Dept.,

IIT Bombay

Lecture–4: Fuzzy

Inferencing

29

th

July 2010Slide2

Meaning of fuzzy subset

Suppose, following classical set theory we say

if Consider the n-hyperspace representation of A and B

(1,1)

(1,0)

(0,0)

(0,1)

x

1

x

2

A

.

B

1

.

B

2

.

B3

Region where Slide3

This effectively means

CRISPLY

P(A) = Power set of AEg: Suppose A = {0,1,0,1,0,1,…………….,0,1} – 10

4 elements

B = {0,0,0,1,0,1,……………….,0,1} – 104 elements

Isn’t with a degree? (only differs in the 2nd element) Slide4

Subset operator is the “odd man” outAUB, A∩B, Ac

are all “Set Constructors” while

A  B is a Boolean Expression or predicate.According to classical logicIn Crisp Set theory A  B is defined as

x xA  xBSo, in fuzzy set theory A

 B can be defined as x µ

A(x) 

µB(x)Slide5

Zadeh’s definition of subsethood goes against the grain of fuzziness theory

Another way of defining A

 B is as follows: x µ

A(x)  µ

B(x)

But, these two definitions imply that µP(B)

(A)=1 where P(B) is the power set of B

Thus, these two definitions violate the fuzzy principle that every belongingness except Universe is fuzzySlide6

Fuzzy definition of subset

Measured in terms of “fit violation”, i.e. violating the condition

Degree of subset hood S(A,B)= 1- degree of superset

=

m(B)

= cardinality of B

= Slide7

We can show that

Exercise 1:

Show the relationship between entropy and subset hoodExercise 2:Prove that

Subset hood of B in ASlide8

Fuzzy sets to fuzzy logic

Forms the foundation of fuzzy rule based system or fuzzy expert system

Expert SystemRules are of the formIf

then

AiWhere

Cis are conditions

Eg: C1

=Colour of the eye yellowC2

= has feverC

3=high bilurubinA = hepatitis Slide9

In fuzzy logic we have fuzzy predicates

Classical logic

P(x1,x2,x3…..x

n) = 0/1Fuzzy Logic

P(x1,x

2,x3

…..xn) = [0,1]

Fuzzy OR

Fuzzy AND

Fuzzy NOTSlide10

Fuzzy ImplicationMany theories have been advanced and many expressions exist

The most used is Lukasiewitz formula

t(P) = truth value of a proposition/predicate. In fuzzy logic t(P) = [0,1]t( ) = min[1,1 -t(P)+t(Q)]

Lukasiewitz definition of implicationSlide11

Linguistic VariablesFuzzy sets are named by Linguistic Variables (typically adjectives).Underlying the LV is a numerical quantity

E.g. For ‘tall’ (LV), ‘height’ is numerical quantity.

Profile of a LV is the plot shown in the figure shown alongside.

μ

tall

(h)

1 2 3 4 5 6

0

height

h

1

0.4

4.5Slide12

Example Profiles

μ

rich

(w)

wealth w

μ

poor

(w)

wealth wSlide13

Example Profiles

μ

A

(x)

x

μ

A

(x)

x

Profile representing

moderate (

e.g.

moderately rich)

Profile representing

extremeSlide14

Concept of HedgeHedge is an intensifierExample:

LV = tall, LV

1 = very tall, LV2 = somewhat tall‘very’ operation: μ

very tall(x) =

μ2

tall(x)‘somewhat’ operation:

μ

somewhat tall(x) =

√(μ

tall(x))

1

0

h

μ

tall

(h)

somewhat tall

tall

very tallSlide15

Fuzzy InferencingTwo methods of inferencing in classical logicModus Ponens

Given

p and pq, infer qModus TolensGiven ~q and p

q, infer ~pHow is fuzzy inferencing done?Slide16

A look at reasoningDeduction: p, p

q

|- qInduction: p1, p2, p3, …|- for_all p

Abduction: q, pq|- p

Default reasoning: Non-monotonic reasoning: Negation by failureIf something cannot be proven, its negation is asserted to be trueE.g., in PrologSlide17

Fuzzy Modus Ponens in terms of truth valuesGiven t(p)=1 and

t(

pq)=1, infer t(q)=1In fuzzy logic, given

t(p)>=a, 0<=a<=1and t(p>q)=c, 0<=c<=1What is

t(q)How much of truth is transferred over the channel

p

qSlide18

Lukasiewitz formulafor Fuzzy Implication

t(P) = truth value of a proposition/predicate. In fuzzy logic t(P) = [0,1]

t( ) = min[1,1 -t(P)+t(Q)]

Lukasiewitz definition of implicationSlide19

Use Lukasiewitz definition

t(

pq) = min[1,1 -t(p)+t(q)]We have t(p->q)=c, i.e., min[1,1 -t(p)+t(q)]=c

Case 1:c=1

gives 1 -t(p)+t(q)>=1, i.e., t(q)>=aOtherwise,

1 -t(p)+t(q)=c, i.e., t(q)>=c+a-1Combining,

t(q)=max(0,a+c-1)This is the amount of truth transferred over the channel

pqSlide20

Two equations consistent

These two equations are consistent with each otherSlide21

ProofLet us consider two crisp sets A and B

1

U

A

B

2

U

B

A

3

U

A

B

4

U

A

BSlide22

Proof (contd…)Case I:So,Slide23

Proof (contd…)Thus, in case I these two equations are consistent with each other

Prove them for other three casesSlide24

Eg: If pressure is high then Volume is low

Pressure/Volume

High Pressure

ANDING of Clauses on the LHS of implication

Low VolumeSlide25

An Example

Controlling

an inverted pendulum:

θ

= angular velocity

Motor

i=currentSlide26

The goal: To keep the pendulum in vertical position (θ

=0)

in dynamic equilibrium. Whenever the pendulum departs from vertical, a torque is produced by sending a current ‘i’Controlling factors for appropriate currentAngle θ

, Angular velocity θ.

Some intuitive rules

If θ is +ve small and

θ.

is –ve smallthen current is zero

If θ is +ve small and

θ. is +ve small

then current is –ve mediumSlide27

-ve med

-ve small

Zero

+ve small

+ve med

-ve med

-ve small

Zero

+ve small

+ve med

+ve med

+ve small

-ve small

-ve med

-ve small

+ve small

Zero

Zero

Zero

Region of interest

Control Matrix

θ

.

θSlide28

Each cell is a rule of the form

If

θ is <> and θ. is <>

then i is <>4 “Centre rules”

if θ = = Zero and

θ.

= = Zero then i = Zeroif

θ is +ve small and θ

. = = Zero then i is –ve small

if θ is –ve small and θ

.= = Zero then i is +ve smallif θ

= = Zero and θ. is

+ve small then i is –ve small

if θ = = Zero and θ

. is –ve small then i is +ve smallSlide29

Linguistic variables

Zero

+ve small-ve smallProfiles

-

ε

+

ε

ε

2

-

ε

2

-

ε

3

ε

3

+ve small

-ve small

1

Quantity (

θ

,

θ

.

, i)

zeroSlide30

Inference procedureRead actual numerical values of

θ

and θ.Get the corresponding

μ values

μZero,

μ(+ve small)

, μ

(-ve small). This is called FUZZIFICATION

For different rules, get the fuzzy I-values from the R.H.S of the rules.“

Collate” by some method and get ONE

current value. This is called DEFUZZIFICATIONResult is one numerical value of

‘i’.Slide31

if θ is Zero and dθ/dt is Zero then i is Zero

if

θ is Zero and dθ/dt is +ve small then i is –ve smallif θ is +ve small and dθ/dt is Zero then i is –ve smallif θ +ve small and dθ

/dt is +ve small then i is -ve medium

-

ε

+

ε

ε

2

-

ε

2

-

ε

3

ε

3

+ve small

-ve small

1

Quantity (

θ

,

θ

.

, i)

zero

Rules InvolvedSlide32

Suppose θ is 1 radian and dθ/dt is 1 rad/sec

μ

zero(θ =1)=0.8 (say)Μ+ve-small(θ =1)=0.4 (say)μzero(dθ/dt =1)=0.3 (say)

μ+ve-small(dθ/dt =1)=0.7 (say)

-

ε

+

ε

ε

2

-

ε

2

-

ε

3

ε

3

+ve small

-ve small

1

Quantity (

θ

,

θ

.

, i)

zero

Fuzzification

1rad

1 rad/secSlide33

Suppose θ is 1 radian and dθ/dt is 1 rad/sec

μ

zero(θ =1)=0.8 (say)μ +ve-small(θ =1)=0.4 (say)μzero(dθ/dt =1)=0.3 (say)

μ+ve-small(dθ/dt =1)=0.7 (say)

Fuzzification

if

θ

is Zero and dθ/dt is Zero then i is Zero

min(0.8, 0.3)=0.3

hence μzero

(i)=0.3if θ is Zero and dθ/dt is +ve small then i is –ve small

min(0.8, 0.7)=0.7 hence

μ-ve-small(i)=0.7

if θ is +ve small and dθ/dt is Zero then i is –ve small

min(0.4, 0.3)=0.3 hence

μ-ve-small(i)=0.3if θ +ve small and dθ/dt is +ve small then i is -ve medium

min(0.4, 0.7)=0.4 hence

μ-ve-medium

(i)=0.4Slide34

-

ε

+

ε

-

ε

2

-

ε

3

-ve small

1

zero

Finding

i

0.4

0.3

Possible candidates:

i=0.5 and -0.5 from the “zero” profile and

μ

=0.3

i=-0.1 and -2.5 from the “-ve-small” profile and

μ

=0.3

i=-1.7 and -4.1 from the “-ve-small” profile and

μ

=0.3

-4.1

-2.5

-ve small

-ve medium

0.7Slide35

-

ε

+

ε

-ve small

zero

Defuzzification: Finding

i

by the

centroid

method

Possible candidates:

i is the x-coord of the centroid of the areas given by the

blue trapezium

, the

green trapeziums

and the black trapezium

-4.1

-2.5

-ve medium

Required i value

Centroid of three

trapezoids