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 1 2 Jesus E. Hernandez and Jaime Nava JJ II
1Department of Electrical and J I Computer Engineering 2Department of Computer Science Page1 of 13
University of Texas at El Paso Go Back El Paso, TX 79968 [email protected] Full Screen [email protected] 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” JJ II
is the one most frequently used as a basic operation. J I
• Fact: “Exclusive or” is also used in commonsense and Page2 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 fop(a, b). Fuzzy “Exclusive Or” . . . • For op = & , we get an “and”-operation (t-norm). Home Page • For op = ∨, we get an “or”-operation (t-conorm). Title Page • As usual, the fuzzy “exclusive or” operation must be JJ II an extension of the corresponding crisp operation ⊕. J I • In the traditional 2-valued logic, 0 ⊕ 0 = 1 ⊕ 1 = 0 and Page3 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 . . . – on a scale from 0 to 8, whatever we pick, we cannot Fuzzy “Exclusive Or” . . . 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 d0 = 6/8 = 0.75 ≈ 0.8. JJ II • It is desirable: that the result of the fuzzy operation J I
not change much if we slightly change the inputs: Page4 of 13 0 0 0 0 |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) JJ II f∨(a, b) = max(a, b). J I • What we do in this presentation: we describe the least Page5 of 13 sensitive fuzzy “exclusive or” operation. Go Back
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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. JJ II • Fact: The weaker the condition, the larger the class of J I operations that satisfy these conditions. Page6 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
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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 . . . • We say that a function f ∈ F is the least sensitive in Main Result the class F if it satisfies the following two conditions: Interpretation of the . . . Fuzzy “Exclusive Or” . . .
– for some real number k, the function f satisfies the Home Page condition Title Page |f(a, b) − f(a0, b0)| ≤ k · max(|a − a0|, |b − b0|); JJ II
– no other function f ∈ F satisfies this condition. J I
Theorem: In the class of all fuzzy “exclusive or” opera- Page7 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
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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
• Natural idea: Title Page – replace ∨ with the least sensitive “or”-operation JJ II f∨(a, b) = max(a, b), J I
– replace & with the least sensitive “and”-operation Page8 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- ing condition with k = 1: Interpretation of the . . . Fuzzy “Exclusive Or” . . . 0 0 0 0 |f(a, b) − f(a , b )| ≤ k · max(|a − a |, |b − b |); Home Page
– 2nd: that no other “exclusive or” operation satisfies Title Page
this property. JJ II
• 1st condition: let us prove that for every ε > 0, if J I 0 0 |a − a | ≤ ε and |b − b | ≤ ε, then Page9 of 13 0 0 |f⊕(a, b) − f⊕(a , b )| ≤ ε. Go Back
• It is known: that the functions min(a, b), max(a, b), Full Screen and 1 − a satisfy the above condition with k = 1. Close
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9. Proof of the Main Result (cont-d) A Crisp “Exclusive Or” . . .
Need for the Least . . . 0 0 • 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 . . . 0 0 | max(a, b) − max(a , b )| ≤ ε; Main Result
Interpretation of the . . . |(1 − a) − (1 − a0)| ≤ ε; and |(1 − b) − (1 − b0)| ≤ ε. Fuzzy “Exclusive Or” . . . • From the result above, by using the condition for the Home Page
max operation, we conclude that Title Page 0 0 | max(1 − a, 1 − b) − max(1 − a , 1 − b )| ≤ ε. JJ II
• Now, from the results above, by using the condition for J I
the min operation, we conclude that Page 10 of 13
| min(max(a, b), max(1 − a, 1 − b)) Go Back − min(max(a0, b0), max(1 − a0, 1 − b0))| ≤ ε. Full Screen • The statement is proven. Close Quit Need for Fuzzy . . .
10. Fuzzy “Exclusive Or” Operations f(a, b) Which A Crisp “Exclusive Or” . . .
Are the Least Sensitive on Average Need for the Least . . .
For t-Norms and t- . . . • Idea: select f so that on average, the change in a and b Definition of a Fuzzy . . . leads to the smallest possible change ∆c in c = f(a, b). Main Result
• Assumption: ∆a and ∆b are independent random vari- Interpretation of the . . . 2 ables with 0 mean and small variance σ . Fuzzy “Exclusive Or” . . . • Objective: estimate ∆c = f(a + ∆a, b + ∆b) − f(a, b). Home Page • Since ∆a and ∆b are small, we can keep only linear Title Page
terms in the Taylor series of ∆c w.r.t. ∆a and ∆b: JJ II ∂f ∂f ∆c ≈ · ∆a + · ∆b. J I ∂a ∂b Page 11 of 13 • Since the variables are independent with 0 mean, the mean of ∆c is also 0, and variance of ∆c is equal to Go Back
! Full Screen ∂f 2 ∂f 2 σ2(a, b) = + · σ2. ∂a ∂b Close
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11. Fuzzy “Exclusive Or” Operations Which Are A Crisp “Exclusive Or” . . .
the Least Sensitive on Average (cont-d) Need for the Least . . .
For t-Norms and t- . . . • Reminder: for each a and b, the variance σ2(a, b) of ∆c Definition of a Fuzzy . . . is equal to Main Result 2 2! ∂f ∂f Interpretation of the . . . σ2(a, b) = + · σ2. ∂a ∂b Fuzzy “Exclusive Or” . . . Home Page • To get the “average” variance, it is reasonable to aver- age this value σ2(a, b) over all possible a and b. Title Page • Resulting average value: I · σ2, where JJ II a=1 b=1 2 2! J I def Z Z ∂f ∂f I = + da db. Page 12 of 13 a=0 b=0 ∂a ∂b Go Back • We want: the average sensitivity to be the smallest. Full Screen • Conclusion: we select the function f(a, b) for which the Close integral I takes the smallest possible value. Quit 12. New Result: Formulation Need for Fuzzy . . . A Crisp “Exclusive Or” . . . • Reminder: we consider “exclusive or” operations f(a, b), Need for the Least . . . For t-Norms and t- . . . i.e., functions f : [0, 1] × [0, 1] → [0, 1] for which: Definition of a Fuzzy . . . Main Result f(0, b) = b, f(a, 0) = a, f(1, b) = 1−b, and f(a, 1) = 1−a. Interpretation of the . . . • Main result: among all such operations, the operation Fuzzy “Exclusive Or” . . .
which is the least sensitive on average has the form Home Page
f⊕(a, b) = a + b − 2 · a · b. Title Page
• Interpretation: JJ II – the classical (2-valued) “exclusive or” operation J I a ⊕ b can be represented as (a ∨ b)&(¬a ∨ ¬b); Page 13 of 13
– use the fuzzy analogues of &, ∨, and ¬ which are Go Back the least sensitive on average: Full Screen
f&(a, b) = max(p+q −1, 0); f∨(a, b) = p+q −p·q; Close
f¬(a) = 1 − a. Quit