1. 16-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 1 CS 621 Artificial Intelligence Lecture 7 - 16/08/05 Prof. Pushpak Bhattacharyya Fuzzy Set (contd) Fuzzy Logic (Start)

2. 16-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 2 Fuzzy Subset U = {1, 2, 3, 4,….,10} A = {1, 2, 3, 4, 5} B = {2, 3, 4} B  A in CRISP SET THEORY  A (x) >=  B (x),  x In terms of membership predicate Crisp subsethood  S1 (x) <=  S2 (x),  x

3. 16-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 3 Geometric Interpretation (0,1) (1,1) (0,0) (1,0) A B1B1 B2B2 B3B3 x2x2 x1x1 U = {x 1, x 2 } B i s are such that  Bi (x) <=  A (x),  x

4. 16-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 4 Geometric Interpretation (Contd 1) The points within the hypercube for which A is the upper right corner are the subsets of A. Space defined by the square is the power set of A. Formulation of ZADEH, classical fuzzy set theory For B to be a subset of A,  B (x) <=  A (x),  x. This means B  P(A) crisply. A B

5. 16-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 5 Geometric Interpretation (Contd 2) Each B i is a subset of A to some degree. A B1B1 B2B2 B3B3 Result of Union, Intersection, Complement is a SET Subsethood is a question

6. 16-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 6 Fuzzy Definition of Subsethood S(B,A) = subsethood of B wrt A = 1 – ∑ x max(0,  B (x) –  A (x)) ∑ x  B (x) Question – Can S(B,A) be 0.

7. 16-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 7 Theorem S(B,A) = m(A  B) m(B) m(S) = cardinality of S = ∑ x  S (x)

8. 16-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 8 Proof of the Theorem Proof: RHS= 1 – ∑ x max(0,  B (x) –  A (x)) ∑ x  B (x) = ∑ x  B (x) – ∑ x max(0,  B (x) –  A (x)) m(B)

9. 16-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 9 Proof of the Theorem (Contd) = ∑ x min(  A (x),  B (x)) m(B) = m(A  B) m(B) = LHS

10. 16-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 10 Entropy of Subsethood E(A) = m(A  A c ) m(A  A c ) S(B,A) = m(A  B) m(B) S(A  A c, A  A c ) = m((A  A c )  (A  A c )) m(A  A c ) = m(A  A c ) = E(A) m(A  A c )

11. 16-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 11 Entropy of Fuzzy Set Entropy of fuzzy set is the degree by which A  A c is a subset of A  A c Entropy is a measure by which WHOLE IS A SUBSET of its OWN PART !!! Subsethood in non-classical fuzzy logic is a degree statement. This influences the notion of Implication.

12. 16-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 12 Fuzzy Logic Set TheoryLogic Set S  S (x) S 1  S 2  S1 (x) ν  S2 (x) S 1  S 2  S1 (x) Λ  S2 (x) S 1  S 2  S1 (x)   S2 (x)

13. 16-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 13 Definitions of Logic Operations Let P 1 and P 2 be fuzzy logic variables /predicates. 0 <= t(P 1 ) <= 1 0 <= t(P 2 ) <= 1Fuzzy Logic

14. 16-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 14 Fuzzy Operations Fuzzy ν : max (t(P 1 ), t(P 2 )) Fuzzy Λ : min(t(P 1 ), t(P 2 )) Fuzzy ~ : 1 – t(P)

15. 16-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 15 Implication LUKISEWITZ LOGIC Many multi-valued logic in 1930 t(P 1 )  t(P 2 ) = min (1, 1-t(P 1 ) + t(P 2 ))

16. 16-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 16 Inferencing Modus Ponens Given P 1 & P 1  P 2 conclude P 2 t(P 1 ) = 1, t(P 1  P 2 ) = 1 conclude t(P 2 ) = 1 - classical logic

17. 16-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 17 Modus Tolens Given ~P 2 and P 1  P 2 conclude ~P 1 i.e t(P2) = 0, t(P 1  P 2 ) = 1 Conclude t(P 2 ) = 0 - classical logic

18. 16-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 18 In Fuzzy Logic We are given t(P 1 ) = a, 0<= a <= 1 t(P 1  P 2 ) = b, 0<= b <=1 What can we say for t(P 2 ) t(P 1  P 2 ) =min(1, 1 – t(P 1 ) + t(P 2 )) By definition Luk. system of logic From given values t(P 1  P 2 ) = min(1, 1 – a + t(P 2 )) t(P 1  P 2 ) = b

19. 16-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 19 Case 1 b = min(1, 1 – a + t(P 2 )) b = 1 1 – a + t(P 2 ) >= 1 or t(P 2 ) >= a - case of complete truth transfer

20. 16-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 20 Case 2 b < 1 1 – a + t(P 2 ) = b or t(P 2 ) = a + b – 1 Combining 1 and 2 t(P 2 ) = a + b -1 But this allows t(P 2 ) to be < 0 t(P 2 ) = max(0, a + b -1) Fuzzy modus ponens.

21. 16-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 21 Fuzzy Modus Tolens t(P 1  P 2 ) = b t(P 2 ) <= a, 0<=a<=1 What is t(P 1 ) Exercise: Deduce expression for fuzzy modus tolens