Need for Fuzzy . . . A Crisp “Exclusive Or” . . . Need for the Least . . . Least Sensitive For t-Norms and t- . . . Definition of a Fuzzy . . . (Most Robust) Main Result Interpretation of the . . . Fuzzy “Exclusive Or” Fuzzy “Exclusive Or” . . . Operations Home Page Title Page Jesus E. Hernandez 1 and Jaime Nava 2 ◭◭ ◮◮ 1 Department of Electrical and ◭ ◮ Computer Engineering Page 1 of 13 2 Department of Computer Science University of Texas at El Paso Go Back El Paso, TX 79968 1 jehernandez7@miners.utep.edu Full Screen 2 jenava@miners.utep.edu Close Quit
Need for Fuzzy . . . 1. Need for Fuzzy “Exclusive Or” Operations A Crisp “Exclusive Or” . . . Need for the Least . . . • One of the main objectives of fuzzy logic is to formalize For t-Norms and t- . . . commonsense and expert reasoning. Definition of a Fuzzy . . . • People use logical connectives like “and” and “or”. Main Result • Commonsense “or” can mean both “inclusive or” and Interpretation of the . . . “exclusive or”. Fuzzy “Exclusive Or” . . . Home Page • Example: A vending machine can produce either a coke or a diet coke, but not both. Title Page ◭◭ ◮◮ • In mathematics and computer science, “inclusive or” is the one most frequently used as a basic operation. ◭ ◮ • Fact: “Exclusive or” is also used in commonsense and Page 2 of 13 expert reasoning. Go Back • Thus: There is a practical need for a fuzzy version. Full Screen • Comment: “exclusive or” is actively used in computer Close design and in quantum computing algorithms Quit
Need for Fuzzy . . . 2. A Crisp “Exclusive Or” Operation: A Reminder A Crisp “Exclusive Or” . . . Need for the Least . . . • Fuzzy analogue of a classical logic operation op: For t-Norms and t- . . . – we know the experts’ degree of belief a = d ( A ) and Definition of a Fuzzy . . . b = d ( B ) in statements A and B ; Main Result – based on a and b , we want to estimate the degree Interpretation of the . . . of belief in “ A op B ”, as f op ( a, b ). Fuzzy “Exclusive Or” . . . Home Page • For op = & , we get an “and”-operation (t-norm). Title Page • For op = ∨ , we get an “or”-operation (t-conorm). ◭◭ ◮◮ • As usual, the fuzzy “exclusive or” operation must be an extension of the corresponding crisp operation ⊕ . ◭ ◮ • In the traditional 2-valued logic, 0 ⊕ 0 = 1 ⊕ 1 = 0 and Page 3 of 13 0 ⊕ 1 = 1 ⊕ 0 = 1. Go Back • Thus, the desired fuzzy “exclusive or” operation f ⊕ ( a, b ) Full Screen must satisfy the same properties: Close f ⊕ (0 , 0) = f ⊕ (1 , 1) = 0; f ⊕ (0 , 1) = f ⊕ (1 , 0) = 1 . Quit
Need for Fuzzy . . . 3. Need for the Least Sensitivity: Reminder A Crisp “Exclusive Or” . . . Need for the Least . . . • One of the main ways to elicit degree of certainty d is For t-Norms and t- . . . to ask to pick a value on a scale. Example: Definition of a Fuzzy . . . – on a scale of 0 to 10, an expert picks 8, so we get Main Result d = 8 / 10 = 0 . 8; Interpretation of the . . . Fuzzy “Exclusive Or” . . . – on a scale from 0 to 8, whatever we pick, we cannot Home Page get 0.8: 6 / 8 = 0 . 75 < 0 . 8; 7 / 8 = 0 . 875 > 0 . 8. – the expert will probably pick 6, with Title Page d ′ = 6 / 8 = 0 . 75 ≈ 0 . 8 . ◭◭ ◮◮ • It is desirable: that the result of the fuzzy operation ◭ ◮ not change much if we slightly change the inputs: Page 4 of 13 | f ( a, b ) − f ( a ′ , b ′ ) | ≤ k · max( | a − a ′ | , | b − b ′ | ) , Go Back with the smallest possible k . Full Screen • Such operations are called the least sensitive or the Close most robust . Quit
Need for Fuzzy . . . 4. For t-Norms and t-Conorms, the Least Sensi- A Crisp “Exclusive Or” . . . tivity Requirement Leads to Reasonable Oper- Need for the Least . . . ations For t-Norms and t- . . . Definition of a Fuzzy . . . • Known results: Main Result – There is only one least sensitive t-norm (“and”- Interpretation of the . . . operation) Fuzzy “Exclusive Or” . . . f & ( a, b ) = min( a, b ) . Home Page – There is also only one least sensitive t-conorm (“or”- Title Page operation) ◭◭ ◮◮ f ∨ ( a, b ) = max( a, b ) . ◭ ◮ • What we do in this presentation: we describe the least Page 5 of 13 sensitive fuzzy “exclusive or” operation. Go Back Full Screen Close Quit
Need for Fuzzy . . . 5. Definition of a Fuzzy Exclusive-Or Operation A Crisp “Exclusive Or” . . . Need for the Least . . . • Definition: A function f : [0 , 1] × [0 , 1] → [0 , 1] is For t-Norms and t- . . . called a fuzzy “exclusive or” operation if Definition of a Fuzzy . . . f (0 , 0) = f (1 , 1) = 0 and f (0 , 1) = f (1 , 0) = 1 . Main Result Interpretation of the . . . • Comment: We could also require other conditions, e.g., Fuzzy “Exclusive Or” . . . commutativity and associativity. Home Page • However, our main objective is to select a single oper- Title Page ation which is the least sensitive. ◭◭ ◮◮ • Fact: The weaker the condition, the larger the class of ◭ ◮ operations that satisfy these conditions. Page 6 of 13 • Thus: the stronger the result that our operation is the Go Back least sensitive in this class. Full Screen • Conclusion: We select the weakest possible condition to make our result as strong as possible. Close Quit
Need for Fuzzy . . . 6. Main Result A Crisp “Exclusive Or” . . . Need for the Least . . . Definition: For t-Norms and t- . . . • Let F be a class of functions from [0 , 1] × [0 , 1] to [0 , 1] . Definition of a Fuzzy . . . Main Result • We say that a function f ∈ F is the least sensitive in Interpretation of the . . . the class F if it satisfies the following two conditions: Fuzzy “Exclusive Or” . . . – for some real number k , the function f satisfies the Home Page condition Title Page | f ( a, b ) − f ( a ′ , b ′ ) | ≤ k · max( | a − a ′ | , | b − b ′ | ); ◭◭ ◮◮ – no other function f ∈ F satisfies this condition. ◭ ◮ Theorem: In the class of all fuzzy “exclusive or” opera- Page 7 of 13 tions, the following function is the least sensitive: Go Back f ⊕ ( a, b ) = min(max( a, b ) , max(1 − a, 1 − b )) . Full Screen Close Quit
Need for Fuzzy . . . 7. Interpretation of the Main Result A Crisp “Exclusive Or” . . . Need for the Least . . . • Reminder: the least sensitive operation is For t-Norms and t- . . . f ⊕ ( a, b ) = min(max( a, b ) , max(1 − a, 1 − b )) . Definition of a Fuzzy . . . Main Result • Fact: in 2-valued logic, “exclusive or” ⊕ can be de- Interpretation of the . . . scribed in terms of the “inclusive or” operation ∨ as Fuzzy “Exclusive Or” . . . a ⊕ b ⇔ ( a ∨ b ) & ¬ ( a & b ) . Home Page Title Page • Natural idea: ◭◭ ◮◮ – replace ∨ with the least sensitive “or”-operation f ∨ ( a, b ) = max( a, b ), ◭ ◮ – replace & with the least sensitive “and”-operation Page 8 of 13 f & ( a, b ) = min( a, b ), and Go Back – replace ¬ with the least sensitive negation opera- Full Screen tion f ¬ ( a ) = 1 − a , Close • Result: we get the expression given in the Theorem. Quit
Need for Fuzzy . . . 8. Proof of the Main Result: 1st Condition A Crisp “Exclusive Or” . . . Need for the Least . . . • Reminder: f ⊕ ( a, b ) = min(max( a, b ) , max(1 − a, 1 − b )). For t-Norms and t- . . . • We need to prove the following two conditions: Definition of a Fuzzy . . . Main Result – 1st: that this function f ⊕ ( a, b ) satisfies the follow- Interpretation of the . . . ing condition with k = 1: Fuzzy “Exclusive Or” . . . | f ( a, b ) − f ( a ′ , b ′ ) | ≤ k · max( | a − a ′ | , | b − b ′ | ); Home Page Title Page – 2nd: that no other “exclusive or” operation satisfies this property. ◭◭ ◮◮ • 1st condition: let us prove that for every ε > 0, if ◭ ◮ | a − a ′ | ≤ ε and | b − b ′ | ≤ ε , then Page 9 of 13 | f ⊕ ( a, b ) − f ⊕ ( a ′ , b ′ ) | ≤ ε. Go Back Full Screen • It is known: that the functions min( a, b ), max( a, b ), and 1 − a satisfy the above condition with k = 1. Close Quit
Need for Fuzzy . . . 9. Proof of the Main Result (cont-d) A Crisp “Exclusive Or” . . . Need for the Least . . . • Known results: if | a − a ′ | ≤ ε and | b − b ′ | ≤ ε , then the For t-Norms and t- . . . following three inequalities hold: Definition of a Fuzzy . . . | max( a, b ) − max( a ′ , b ′ ) | ≤ ε ; Main Result Interpretation of the . . . | (1 − a ) − (1 − a ′ ) | ≤ ε ; and | (1 − b ) − (1 − b ′ ) | ≤ ε. Fuzzy “Exclusive Or” . . . • From the result above, by using the condition for the Home Page max operation, we conclude that Title Page | max(1 − a, 1 − b ) − max(1 − a ′ , 1 − b ′ ) | ≤ ε. ◭◭ ◮◮ ◭ ◮ • Now, from the results above, by using the condition for the min operation, we conclude that Page 10 of 13 | min(max( a, b ) , max(1 − a, 1 − b )) Go Back − min(max( a ′ , b ′ ) , max(1 − a ′ , 1 − b ′ )) | ≤ ε. Full Screen Close • The statement is proven. Quit
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