Soft Computing Fuzzy Propositions Prof. Debasis Samanta Department of Computer Science & Engineering IIT Kharagpur
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Fuzzy Propositions • Two-valued logic vs. Multi-valued logic • Examples of Fuzzy proposition • Fuzzy proposition vs. Crisp proposition • Canonical representation of Fuzzy proposition • Graphical interpretation of Fuzzy proposition
Debasis Samanta CSE IIT Kharagpur
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Two-valued logic vs. Multi-valued logic • The basic assumption upon which crisp logic is based - that every proposition is either TRUE or FALSE. • The classical two-valued logic can be extended to multi-valued logic. • As an example, three valued logic to denote true(1), false(0) and indeterminacy ( 1/2 ).
Debasis Samanta CSE IIT Kharagpur
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Two-valued logic vs. Multi-valued logic Different operations with three-valued logic can be extended as shown in the truth table: a b 0
0
0
0
1
1
1
0
½
0
½
1
1
½
0
1
0
1
1
1
0
½
0
0
½
½
½
½
½
½
½
½
½
½
1
½
1
½
1
½
1
½
1
0
0
1
0
0
0
1
½
½
1
0
½
½
1
1
1
1
0
1
1 Debasis Samanta CSE IIT Kharagpur
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Three-valued logic Fuzzy connectives defined for such a three-valued logic better can be stated as follows: Symbol
: Absolutely false : Partially false : May be false or not false : May be true or not true : Partially true : Absolutely true.
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Fuzzy proposition: Example 2
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Fuzzy proposition: Example 2
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Fuzzy proposition vs. Crisp proposition • The fundamental difference between crisp (classical) proposition and fuzzy propositions is in the range of their truth values. • While each classical proposition is required to be either true or false, the truth or falsity of fuzzy proposition is a matter of degree. • The degree of truth of each fuzzy proposition is expressed by a value in the interval [0,1] both inclusive. Debasis Samanta CSE IIT Kharagpur
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Canonical representation of Fuzzy proposition
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Canonical representation of Fuzzy proposition
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Graphical interpretation of fuzzy proposition
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Fuzzy system
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Thank You!! Debasis Samanta CSE IIT Kharagpur
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Soft Computing Fuzzy Implication Prof. Debasis Samanta Department of Computer Science & Engineering IIT Kharagpur
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Fuzzy implications • Fuzzy rule • Examples of fuzzy implications • Interpretation of fuzzy rules • Product operators • Zadeh’s Max-Min rule and some examples
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Fuzzy rule
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Fuzzy implication : Example 1
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Fuzzy implication : Example 2
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Fuzzy implication : Example 2
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Interpretation of fuzzy rules
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Interpretation as A coupled with B
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Interpretation as A coupled with B
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Interpretation as A coupled with B
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Product Operators
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Interpretation of A entails B
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Interpretation of A entails B
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Interpretation of A entails B
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Example 3: Zadeh’s Max-Min rule
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Example 3: Zadeh’s Max-Min rule
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Example 3: Zadeh’s Max-Min rule
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Example 4:
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Example 4:
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Example 4:
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Example 4:
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Example 4:
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Thank You! 23
Soft Computing Fuzzy Inferences Debasis Samanta Department of Computer Science and Engineering IIT KHARAGPUR
Fuzzy inferences
An example from propositional logic
Inferring procedures in Fuzzy logic
Fuzzy inferring procedures
Generalized Modus Ponens : Example
Example: Generalized Modus Ponens
Generalized Modus Ponens
Example. Generalized Modus Ponens
Example. Generalized Modus Ponens
Example. Generalized Modus Tollens
Example. Generalized Modus Tollens
Example. Generalized Modus Tollens
Practical example
Practice
Soft Computing Defuzzyfication Techniques-I Debasis Samanta Department of Computer Science and Engineering IIT KHARAGPUR
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What is defuzzification?
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Example 2. Fuzzy to crisp As an another example, let us consider a fuzzy set whose membership function is shown in the following figure.
What is the crisp value of the fuzzy set in this case? Debasis Samanta CSE IIT Kharagpur
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Example 3. Fuzzy to crisp
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Why defuzzification?
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Generic structure of a Fuzzy system Following figure shows a general framework of a fuzzy system.
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Defuzzification Techniques
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Defuzzification methods A number of defuzzification methods are known. Such as 1)
Lambda-cut method
2)
Weighted average method
3)
Maxima methods
4)
Centroid methods
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Lambda-cut method
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Lambda-cut method Lambda-cut method is applicable to derive crisp value of a fuzzy set or relation. • Lambda-cut method for fuzzy relation The same has been applied to Fuzzy set • Lambda-cut method for fuzzy set
In many literature, Lambda-cut method is also alternatively termed as Alpha-cut method.
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Lamda-cut method for fuzzy set
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Lambda-cut for a fuzzy set : Example
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Lambda-cut sets : Example Two fuzzy sets P and Q are defined on x as follows.
P
0.1
0.2
0.7
0.5
0.4
Q
0.9
0.6
0.3
0.2
0.8
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Lambda-cut for a fuzzy relation
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Some properties of -cut relations
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Summary: Lambda-cut methods
Lambda-cut method converts a fuzzy set (or a fuzzy relation) into a crisp set (or relation).
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Output of a Fuzzy System
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Output of a fuzzy System
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Output fuzzy set : Illustration
For instance, let us consider the following:
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Output fuzzy set : Illustration
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Output fuzzy set : Illustration
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Output fuzzy set : Illustration
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Thank You!! 24
Soft Computing Defuzzyfication Techniques-II Debasis Samanta Department of Computer Science and Engineering IIT KHARAGPUR
Page 2 of 33. 2. ASIMAVA ROY CHOUDHURY. MECHANICAL ENGINEERING. IIT KHARAGPUR. A cutting tool is susceptible to breakage, dulling and wear. TOOL WEAR AND TOOL LIFE. Rake. surface. Pr. flank. Aux. flank. Page 2 of 33. Page 3 of 33. 3. ASIMAVA ROY CHOU
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