CS 621 Artificial Intelligence Lecture 6 – 09/08/05 Prof. Pushpak Bhattacharyya

09-08-05 Prof. Pushpak Bhattacharyya, IIT Bombay

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CS 621 Artificial Intelligence

Lecture 6 – 09/08/05

Prof. Pushpak Bhattacharyya

FUZZY SET THEORY (Contd)


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Digression Recap

S

G

Path 1

Path 2both optimal

A1*, A

2*

A1* is less informed

h1(n) < h

2(n) n


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Example

S

A B C

G

7 6 5

543

A1*

h1(n) = 2, n

A2*

h2(n) = 3, n


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Status of Lists in A2*

A2* --> 1. OL: S

CL: Φ2. OL: A B C (f

A=10, f

B=9, f

C=8)

CL: S3. OL: A(10), B(9), g(f=10)

CL: S C4. OL: A G

CL: S C B5. OL: A

CL: S C B G


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Status of Lists in A1*

A1* --> 1. OL: S

CL: Φ2. OL: A B C (f

A=9, f

B=9, f

C=7)

CL: S3. OL: A(9), B(8), g(f=10)

CL: S C


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Definitions Seen Till Now

Membership Predicate : 0 <= µS(x) <= 1

Profile : 2 dimension figure of µ and its valueHedge : deals with adverbLinguistic Variable


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Fuzzy Set Operations

Union : A, B : fuzzy setsµ

AυB(x) = max(µ

A(x), µ

B(x))

Intersection : µA∩B

(x) = min(µA(x), µ

B(x))

Complementation : µAc(x) = 1 - µ

A(x)


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Laws of Crisp Set Theory Apply

1. (Ac)c = A

µ(A

c)c (x) = 1 - µ

Ac (x)

= 1 – (1 - µA(x))

= µA(x) QED

2. A υ Ac = Ac υ A

3. A ∩ Ac = Ac ∩ A

4. De Morgan(A υ B)c = Ac ∩ Bc

(A ∩ B)c = Ac υ Bc


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More Laws

5. AssociativityA υ B υ C = (A υ B) υ C = A υ (B υ C)A ∩ B ∩ C = (A ∩ B) ∩ C = A ∩ (B ∩ C)

6. DistributivityA ∩ (B υ C) = (A ∩ B) υ (A ∩ C)


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Proof

Prove (A ∩ B)c = Ac υ Bc

LHS: = µ(A∩B)

c(x)

= 1 - µ(A∩B)

(x)

= 1 - min(µA(x) , µ

B(x))

= max (1 - µA(x) , 1 - µ

B(x))

= RHS


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Representation of Fuzzy SetsX = {x

1,x

2} UNIVERSE

CRISP SUBSETS of X are Φ, {x1}, {x

2}, {x

1, x

2}

x2

(0,0) (1,0)

(1,1)(0,1)

UNIT SQUARE

x1

● Sets can be represented by membership values.● Given n-element UNIVERSE, the subsets of the UNIVERSE correspond to the CORNERS of the HYPERCUBE which is UNIT● Fuzzy subsets of U are the points within the UNIT HYPERCUBE● Infinite number of fuzzy subsets


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Representation (Contd)

A fuzzy set S is represented by

S = {µS(x

1)/x

1, µ

S(x

2)/x

2 , .... , µ

S(x

n)/x

n}

{x2}

(0,0) (1,0)

(1,1)(0,1)

x1

Φ {x1}

{x1, x

2}

A(.3, .5) Ac

A = {.3/x1, .5/x

2}


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Unit Square

{x2}

(0,0) (1,0)

(1,1)(0,1)

x1

Φ {x1}

{x1, x

2}

A

Ac

A∩Ac

AυAc

Corners : A, Ac, A∩Ac, AυAc

Fuzziness decreases as we move towards the corners. At the centre, fuzziness is maximum.


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Unit Square (Contd)

Centre corresponds to “Russel's Paradox”

Statement: A barber in a city shaves only and all those who do not shave themselves.

Ques: Does the barber shave himself?

P → Barber shaves himselfP → ~Palso ~P → Pso, P ≡ ~Pt(P) = 1 – t(P)

so, t(P) = 0.5


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Degree of Fuzziness

A point nearer to the corner is less fuzzy.

Measure of fuzziness = Entropy of fuzzy set

Entropy = E(A)= d(A, A

near) / d(A, A

far)

Anear

= point closest to A out of the 4 corners

Afar

= farthest corner point


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Degree (Contd)

d = distance between A, B

= ∑x | µ

A(x) - µ

B(x) |

A = {0.3/x1, 0.7/x

2}

B = {0.8/x1, 0.1/x

2}

d(A,B) = 0.5 + 0.6 = 1.1

L1 - distance


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Entropy

x2

(0,0) (1,0)

(1,1)(0,1)

x1

{x1, x

2}

A

B

E(Acentre

) = 1

0 <= E(A) <= 1

Theorem: E(A) = m(A ∩ Ac) / m(A υAc)

m(A) = “cardinality” of A= Σ

x µ

A(x)


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Proof Sketch

From definition

E(A) = d(A, Anear

) / d(A, Afar

)

d(A, Anear

) = ∑xi | µA

(xi) - µ

A near(x

i) |

d(A, Afar

) = ∑xi | µA

(xi) - µ

A far(x

i) |

m(A Ac) = ∑xi min(µ

A(x

i), µ

Ac (x

i))

m(A Ac) = ∑xi max(µ

A(x

i), µ

Ac (x

i))


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Fuzziness & Probability

Both model uncertainity.Sum of membership values ≠ 1 necessarily.

Entropy(A), relates to probability of an event.


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Summary

● Fuzzy set operations: generalizations of crisp set operations● Geometric representation of fuzzy set● Introduced degree of fuzziness● Measured in terms of entropy.

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CS 621 Artificial Intelligence Lecture 6 – 09/08/05 Prof. Pushpak Bhattacharyya