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Fuzzy Implications

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1. 1. 1 Amity Institute of Applied Sciences FUZZY IMPLICATIONS Summer Project MASTER OF SCIENCE IN APPLIED MATHEMATICS By Himani Jain Enrollment No: A4451014015 Under the Supervision of: Dr. Rashmi Singh
2. 2. 2 In real world, there exists much fuzzy knowledge; Knowledge that is vague, imprecise, uncertain, ambiguous, iexact, or probabilistic in nature. Fuzzy sets have been introduced by Lofti A. Zadeh (1965) as an extension of the classical notion of set Human thinking and reasoning frequently involve fuzzy information, originating from inherently inexact human concepts. Humans, can give satisfactory answers, which are probably true. However, our systems are unable to answer many questions. The reason is, most systems are designed based upon classical set theory and two-valued logic which is unable to cope with unreliable and incomplete information and give expert opinions. Amity Institute of Applied Sciences INTRODUCTION
3. 3. 3 We want, our systems should also be able to cope with unreliable and incomplete information and give expert opinions. Fuzzy sets have been able to provide solutions to many real world problems. Fuzzy set theory is an extension of classical set theory where elements have degrees of membership. Amity Institute of Applied Sciences
4. 4. • The use of linguistic variables and fuzzy propositions in the fuzzy contexts can be an interesting tool to extract a more complete information from them. • Uncertainty -- Arising from imprecision, vagueness, non-specificity, inconsistency ○ Traditional view: -- Uncertainty is undesirable and should be avoided. ○ Modern view: -- Uncertainty is not only an unavoidable plague, but it has a great utility. 4 Amity Institute of Applied Sciences
5. 5. 5 Crisp Sets ○ Methods for describing sets 1. Enumeration 2. Description 3. Characteristic function Amity Institute of Applied Sciences
6. 6. 6 Amity Institute of Applied Sciences
7. 7. 7 Fuzzy Sets: Basic Types ○ Fuzzy sets －Sets with vague boundaries －Membership of x in A is a matter of degree to which x is in A ○ Utilization of fuzzy sets (1) Representation of uncertainty (2) Representation of conceptual entities e.g., expensive, close, greater, sunny, tall Amity Institute of Applied Sciences
8. 8. ○ Fuzzy Sets Crisp Sets membership characteristic function function Amity Institute of Applied Sciences 8
9. 9. 10 ◎ Fuzzy Set Operations -- Standard complement︰ -- Equilibrium points︰ -- Standard intersection︰ -- Standard union: -- Difference Amity Institute of Applied Sciences
10. 10. 11 Amity Institute of Applied Sciences CLASSICAL IMPLICTIONS Classical implications are of two types: conditional and biconditional. Conditional : The conditional p  q, which we read "if p, then q" or "p implies q," is  defined by the following truth table. The arrow “" is the conditional operator, and in p  q the statement p  is  called  the antecedent,  or hypothesis,  and  q  is  called  the consequent, or conclusion. p q p  q T T T T F F F T T F F T
11. 11. 12 Amity Institute of Applied Sciences Biconditional : The biconditional p q, which we read "p if and only if q" or "p is equivalent to q," is defined by the following truth table. ↔ p q p q T T T T F F F T F F F T ↔ The arrow " " is the biconditional operator. Note that, from the truth table, we see that, for p    q to be true, both p and q  must have the same truth values; otherwise it is false. ↔ ↔
12. 12. 13 Amity Institute of Applied Sciences FUZZY IMPLICATION Definition 1 A binary operator �� on [0, 1] is called to be an im plicatio n  functio n (or an  im plicatio n) if it satisfies : i)  �� ሺ�, �ሺ≥ ��(�, �) �ℎ𝑒� � ≤ �, for all � ∈ [0, 1]; ii)  ��(�, �) ≤ �� (�, �) �ℎ𝑒� � ≤ �, for all � ∈ [0, 1]; iii)  �� (0, 0) = �� (1, 1) = �� (0, 1) = 1 and �� (1, 0) = 0 etc.
13. 13. 14 Amity Institute of Applied Sciences Definition 2 We say that the following properties are satisfied by a fuzzy implication ��: 1) The le ft  ne utra lity  pro pe rty (NP), if �� (1, �) = � , � ∈ [0,1] 2)  The o rde ring   pro pe rty (OP), if � ≤ � ՞ ��(�, �) = 1, �, � ∈ [0, 1] 3) The ide ntity  pro pe rty (IP), if �� (�, �) = 1 , �, � [0, 1]
14. 14. 15 Amity Institute of Applied Sciences 4) The e xcha ng e   pro pe rty (EP), if �� (�, �� (�, �)) = �� (�, �� (�, �)) , �, �, � [0, 1] 5) The law  o f  co ntrapo sitio n with respect to a fuzzy negation �, (�(�)), if �� ሺ�, �ሺ= ��(�(�), �(�)) , �, �, � [0, 1] Some basic fuzzy implications are in next slide…
15. 15. 16 Amity Institute of Applied Sciences
16. 16. 17 Amity Institute of Applied Sciences LATTICE ON FUZZY IMPLICATION Definition 1 A partially o rde re d re latio n is a relation ≤ which is defined on any fuzzy implication and satisfies following properties : i) Reflexive. ii) Antisymmetric. iii) Transitive. And (��, ≤) is called a po se t.
17. 17. 18 Amity Institute of Applied Sciences Definition 2 A lattice o f a fuz z y im plicatio n is a partial ordered implication (��, ≤) in which every pair of implication �, � ∈ �� has a greatest lower bound (meet) and least upper bound (join) ∧ {�, �} = 𝑚𝑒𝑒�{�, �} = 𝐺𝑙� {�, �} = {� (�, �) ∩ � (�, �)} = 𝑚�� {�, �} ∨ {�, �} = 𝑗𝑜��{�, �} = �𝑢� {�, �} = {� (�, �) ∪ � (�, �)} = 𝑚𝑎� {�, �} A Fuzzy implication lattice will always be nonempty; but it can consist of only one element - the smallest element coincides with the largest one.
18. 18. 19 Amity Institute of Applied Sciences SOME PROPERTIES OF FUZZY IMPLICATION LATTICE Following are some properties of the two binary operations of max and min on a fuzzy implication lattice (��, ≤) , for L, M, N ∈ �� we have I. Idempotent Law 𝑚�� (� (�, �), � (�, �)) = � (�, �) 𝑚𝑎� (� (�, �), � (�, �)) = � (�, �)
19. 19. 20 Amity Institute of Applied Sciences II. Commutative Law 𝑚�� (� (�, �), � (�, �)) = 𝑚�� (� (�, �), � (�, �)) 𝑚𝑎� (� (�, �), � (�, �)) = 𝑚𝑎� (� (�, �), � (�, �)) III. Associative Law 𝑚��(𝑚��(�(�, �), �(�, �)), �(�, �)) = 𝑚��(�(�, �), 𝑚��(�(�, �), �(�, �))) 𝑚𝑎�(𝑚𝑎�(�(�, �), �(�, �)), �(�, �)) = 𝑚𝑎�(�(�, �), 𝑚𝑎�(�(�, �), �(�, �)))
20. 20. 21 Amity Institute of Applied Sciences IV. Absorption Law 𝑚�� (𝑚𝑎� (�(�, �), �(�, �)), �(�, �)) = �(�, �) 𝑚𝑎� (𝑚�� (�(�, �), �(�, �)), �(�, �)) = �(�, �) In the following slides we show an example of fuzzy implication lattice.
21. 21. 22 Amity Institute of Applied Sciences EXAMPLE : Let (��, ≤) be a fuzzy implication. Their greater lower bound is max and lowest upper bound is min. Let ���ሺ�, �ሺ, � 𝑅�ሺ�, �ሺ, � 𝑊�ሺ�, �ሺ∈ ��. Now we will show that these satisfy all the above four properties where �, � ∈ [0, 1] and NAME FORMULA Lukasiewicz ILK ሺx, yሺ= ( 1, 1 − x + y) Rescher IRS ሺx, yሺ= ሺ 1, if x ≤ y 0, if x > � Weber IWB ሺx, yሺ= ሺ 1, if x < 1 y, if x = 1
22. 22. 23 Amity Institute of Applied Sciences SOLUTION Let x = 0.7 and y = 0.6 ,then ���ሺ0.7, 0.6ሺ= minሺ1, 1 − 0.7 + 0.6ሺ= minሺ1, 0.9ሺ= 0.9 � 𝑅�ሺ0.7, 0.6ሺ= 0 (𝑎� � > �) � 𝑊� ሺ0.7, 0.6ሺ= 1 (𝑎� 0.7 < 1)
23. 23. 24 Amity Institute of Applied Sciences Now, 1) Idempotent Law : • 𝑚�� (���ሺ�, �ሺ, ���(�, �)) = 𝑚��(0.9, 0.9) = 0.9 = ���(�, �) • 𝑚𝑎� (���ሺ�, �ሺ, ���(�, �)) = 𝑚𝑎�(0.9, 0.9) = 0.9 = ���(�, �)
24. 24. 25 Amity Institute of Applied Sciences 2) Commutative Law : • L. H. S = min൫�𝐿�ሺ�, �ሺ, � 𝑊�ሺ�, �ሺ൯= minሺ0.9, 1ሺ = 0.9 = minሺ1, 0.9ሺ= min൫� 𝑊�ሺ�, �ሺ, �𝐿�ሺ�, �ሺ൯= �. �. � • 𝐿. �. � = max൫�𝐿�ሺ�, �ሺ, � 𝑊�ሺ�, �ሺ൯= maxሺ0.9, 1ሺ = 1 = maxሺ1, 0.9ሺ= 𝑚𝑎�൫� 𝑊�ሺ�, �ሺ, �𝐿�ሺ�, �ሺ൯= �. �. �
25. 25. 26 Amity Institute of Applied Sciences P a g e | 13 3) Associative Law : • 𝐿. �. � = min ሺmin൫ILK ሺx, yሺ, IRS ሺx, yሺ൯, IWB ሺx, yሺሺ = minሺminሺ0.9, 0ሺሺ, 1) = minሺ0, 1ሺ = 0 �. �. � = min⁡(ILK ሺx, yሺ, min⁡(IRS ሺx, yሺ, IWB ሺx, yሺ)) = minሺ0.9, minሺ0,1ሺሺ = minሺ0.9, 0ሺ 0
26. 26. 27 Amity Institute of Applied Sciences • L. H. S = max⁡(max൫ILK ሺx, yሺ, IRS ሺx, yሺ൯, IWB ሺx, yሺ) = maxሺmax⁡(0.9, 0ሺ), 1) = maxሺ0.9, 1ሺ = 1 �. �. � = max൫�𝐿�ሺ�, �ሺ, max൫���ሺ�, �ሺ, � 𝑊�ሺ�, �ሺ൯൯ = maxሺ0.9, maxሺ0,1ሺሺ = maxሺ0.9,1ሺ = 1
27. 27. 28 Amity Institute of Applied Sciences P a g e | 15 4)Absorption Law : • L. H. S = min ሺmax൫ILK ሺx, yሺ, IRS ሺx, yሺ൯, IWB ሺx, yሺሺ = minሺmaxሺ0.9,0ሺ, 1ሺ= minሺ0.9,1ሺ= 0.9 = ILK ሺx, yሺ= �. �. � • 𝐿. �. � = max ሺmin൫� 𝐿� ሺ�, �ሺ, ��� ሺ�, �ሺ൯, � 𝑊� ሺ�, �ሺሺ = maxሺminሺ0.9,0ሺ, 1ሺ= maxሺ0,1ሺ= 1 = � 𝑊� ሺ�, �ሺ= �. �. �
28. 28. 29 Amity Institute of Applied Sciences