Outline Fuzzy Logic: Reminder How t-Norms Are . . . In Practice, the . . . Approximate Nature of Ideal Case: Example Traditional Fuzzy What Happens When . . . Natural Idea Leads to . . . Methodology Naturally A Slightly More . . . Complex Numbers Are . . . Leads to Complex-Valued Home Page Fuzzy Degrees Title Page ◭◭ ◮◮ Olga Kosheleva and Vladik Kreinovich ◭ ◮ University of Texas at El Paso, El Paso, TX 79968, USA Page 1 of 13 olgak@utep.edu, vladik@utep.edu Go Back Full Screen Close Quit
Outline Fuzzy Logic: Reminder 1. Outline How t-Norms Are . . . • In the traditional fuzzy logic, the experts’ degrees of In Practice, the . . . confidence are described by numbers from [0 , 1]. Ideal Case: Example What Happens When . . . • These degree have a clear intuitive meaning. Natural Idea Leads to . . . • Surprisingly, in some applications, it is useful to also A Slightly More . . . consider complex-valued degrees. Complex Numbers Are . . . Home Page • The intuitive meaning of complex-valued degrees is not clear. Title Page • In this talk, we provide a possible explanation for the ◭◭ ◮◮ success of complex-valued degrees. ◭ ◮ • We show that these degrees naturally appear due to Page 2 of 13 the approximate nature of fuzzy methodology. Go Back • This explanation makes the use of complex-valued de- Full Screen grees more intuitively understandable. Close Quit
Outline Fuzzy Logic: Reminder 2. Fuzzy Logic: Reminder How t-Norms Are . . . • Experts are not 100% sure about their statements. In Practice, the . . . Ideal Case: Example • To describe the expert’s degree of certainty, fuzzy logic What Happens When . . . uses numbers from the interval [0 , 1]. Natural Idea Leads to . . . • For example, if an expert marks his certainty as 8 on A Slightly More . . . a scale of 0 to 10, we take d = m/n . Complex Numbers Are . . . Home Page • Ideally, we should elicit expert’s degree of confidence in all possible combinations of his/her statements. Title Page • However, there are exponentially many such combina- ◭◭ ◮◮ tions, so we cannot ask the expert about all of them. ◭ ◮ • Thus, we need to estimate d ( A & B ) based on a = d ( A ) Page 3 of 13 and b = d ( B ). Go Back • The resulting estimate f & ( a, b ) is known as an “and”- Full Screen operation (t-norm). Close Quit
Outline Fuzzy Logic: Reminder 3. How t-Norms Are Determined: Reminder How t-Norms Are . . . • We find a t-norm empirically: for several pairs of state- In Practice, the . . . ments ( A k , B k ), Ideal Case: Example What Happens When . . . – we elicit the degrees d ( A k ), d ( B k ), and d ( A k & B k ), Natural Idea Leads to . . . – and then we find f & ( a, b ) for which, for all k , A Slightly More . . . d ( A k & B k ) ≈ f & ( d ( A k ) , d ( B k )) . Complex Numbers Are . . . Home Page • For example, we can use the Least Squares method and Title Page find f & ( a, b ) for which � ◭◭ ◮◮ ( d ( A k & B k ) − f & ( d ( A k ) , d ( B k ))) 2 → min . ◭ ◮ k Page 4 of 13 • This procedure should lead to real-valued degrees. Go Back • Interestingly, sometimes complex-valued degrees are use- Full Screen ful. Close Quit
Outline Fuzzy Logic: Reminder 4. In Practice, the Situation May Be Somewhat How t-Norms Are . . . More Complicated In Practice, the . . . • Sometimes: Ideal Case: Example What Happens When . . . – instead of knowing the expert’s degree of belief in Natural Idea Leads to . . . the basic statements, A Slightly More . . . – we only know the expert’s degree of belief in some Complex Numbers Are . . . propositional combinations of the basic statements. Home Page • In this case: Title Page – first, we need to recover the degrees d 1 , . . . , d n from ◭◭ ◮◮ the available information; ◭ ◮ – then, we use d i to estimate the expert’s degree of belief in other propositional combinations. Page 5 of 13 • Ideal case: d ( A & B ) = f & ( d ( A ) , d ( B )) and d ( A ∨ B ) = Go Back f ∨ ( d ( A ) , d ( B )). Full Screen • In this case, we can recover the desired degrees. Close Quit
Outline Fuzzy Logic: Reminder 5. Ideal Case: Example How t-Norms Are . . . • Example: f & ( a, b ) = a · b , f ∨ ( a, b ) = a + b − a · b , and In Practice, the . . . the actual (unknown) values are d 1 = 0 . 4 and d 2 = 0 . 6. Ideal Case: Example What Happens When . . . • We only know the values d ( S 1 & S 2 ) = 0 . 4 · 0 . 6 = 0 . 24 Natural Idea Leads to . . . and d ( S 1 ∨ S 2 ) = 0 . 6 + 0 . 4 − 0 . 6 · 0 . 4 = 0 . 76 . A Slightly More . . . • To reconstruct d i , we form equations d 1 · d 2 = 0 . 24 and Complex Numbers Are . . . d 1 + d 2 − d 1 · d 2 = 0 . 76. Home Page • Adding these equations, we get d 1 + d 2 = 1, hence Title Page d 2 = 1 − d 1 . ◭◭ ◮◮ • Substituting d 2 = 1 − d 1 into d 1 · d 2 = 0 . 24, we get ◭ ◮ d 2 1 − d 1 + 0 . 24 = 0, hence �� 1 Page 6 of 13 � 2 √ d 1 = 1 Go Back − 0 . 24 = 0 . 5 ± 0 . 25 − 0 . 24 = 0 . 5 ± 0 . 1 . 2 ± 2 Full Screen • Thus, d 1 = 0 . 4 or d 1 = 0 . 6, as expected. Close Quit
Outline Fuzzy Logic: Reminder 6. What Happens When the “And”- and “Or”- How t-Norms Are . . . Operations Are Only Approximate? In Practice, the . . . • Let’s assume that on average, the expert’s reasoning is Ideal Case: Example best described by f & ( a, b ) = a · b , f ∨ ( a, b ) = a + b − a · b . What Happens When . . . Natural Idea Leads to . . . • This does not mean, of course, that we always have A Slightly More . . . d ( A & B ) = f & ( d ( A ) , d ( B )). Complex Numbers Are . . . • For example, when S 2 = S 1 and d ( S 1 ) = 0 . 5, we have Home Page d ( S 1 & S 2 ) = d ( S 1 ∨ S 2 ) = d ( S 1 ) = 0 . 5 � = 0 . 5 · 0 . 5 . Title Page • Let us see what happens if we try to reconstruct d i ◭◭ ◮◮ from d ( S 1 & S 2 ) = 0 . 5 and d ( S 1 ∨ S 2 ) = 0 . 5. ◭ ◮ • From d 1 · d 2 = 0 . 5 and d 1 + d 2 − d 1 · d 2 = 0 . 5 , we get d 2 = 1 − d 1 and d 2 1 − d 1 + 0 . 5 = 0 . Page 7 of 13 • This equation does not have any real solutions, only Go Back complex ones. Full Screen • So, it makes sense to use complex degrees d i . Close Quit
Outline Fuzzy Logic: Reminder 7. Natural Idea Leads to Complex-Valued Degrees How t-Norms Are . . . • From d 2 1 − d 1 + 0 . 5 = 0 , we get d 1 = 0 . 5 ± 0 . 5 · i . In Practice, the . . . Ideal Case: Example • It is difficult to interpret complex-valued degrees. What Happens When . . . • So, it is natural, for each such complex-valued degree, Natural Idea Leads to . . . to take the closest value from the interval [0 , 1]. A Slightly More . . . • For complex numbers, the natural distance is Euclidean Complex Numbers Are . . . � ( a 1 − b 1 ) 2 + ( a 2 − b 2 ) 2 . Home Page distance d ( a 1 + a 2 · i , b 1 + b 2 · i) = Title Page • It is easy to see that for a complex number a 1 + a 2 · i, the closest point on [0 , 1] is: ◭◭ ◮◮ – the value a 1 is a 1 ∈ [0 , 1]; ◭ ◮ – the value 0 is a 1 < 0, and Page 8 of 13 – the value 1 if a 1 > 1. Go Back • Thus, for 0 . 5 ± 0 . 5 · i, the closest number from [0 , 1] is Full Screen 0.5: exactly what the expert assigned! Close Quit
Outline Fuzzy Logic: Reminder 8. A Slightly More General Example How t-Norms Are . . . • Let’s consider the same “and”- and “or”-operations In Practice, the . . . and S 1 = S 2 , but with a general d = d ( S 1 ) = d ( S 2 ) . Ideal Case: Example What Happens When . . . • In this example, we get a system of equations d 1 · d 2 = d Natural Idea Leads to . . . and d 1 + d 2 − d 1 · d 2 = d . A Slightly More . . . • After adding these two equations, we get d 1 + d 2 = 2 d , Complex Numbers Are . . . hence d 2 = 2 d − d 1 . Home Page • Substituting d 2 = 2 d − d 1 into the first equation, we Title Page get d 1 · (2 d − d 1 ) = d and d 2 1 − 2 d · d 1 + d = 0 . ◭◭ ◮◮ √ d − d 2 · i . • Thus, d 1 = d ± ◭ ◮ • For both complex values d i , the closest number from Page 9 of 13 the interval [0 , 1] is the value d . Go Back • This is also exactly what the experts assigned. Full Screen Close Quit
Outline Fuzzy Logic: Reminder 9. Complex Numbers Are Not a Panacea How t-Norms Are . . . • One may get a false impression that complex numbers In Practice, the . . . always lead to perfect results. Ideal Case: Example What Happens When . . . • To avoid this impression, let’s consider another exam- Natural Idea Leads to . . . ple when S 2 implies S 1 . A Slightly More . . . • In this case, S 1 & S 2 is simply equivalent to S 2 , and Complex Numbers Are . . . S 1 ∨ S 2 is equivalent to S 1 . Home Page • So, for example, for d 1 = 0 . 6 and d 2 = 0 . 4, we get Title Page d ( S 1 & S 2 ) = 0 . 4 and d ( S 1 ∨ S 2 ) = 0 . 6 . ◭◭ ◮◮ • In this example, we get a system of equations d 1 · d 2 = ◭ ◮ 0 . 4 and d 1 + d 2 − d 1 · d 2 = 0 . 6. √ Page 10 of 13 • So, d 2 1 − d 1 + 0 . 4 = 0 , and d 1 = 0 . 5 ± 0 . 15 · i . Go Back • For both d 1 , the closest number from [0 , 1] is 0 . 5. Full Screen • This is different from 0.4 and 0.6 – though close. Close Quit
Recommend
More recommend
