The Origins of Fuzzy . . . Traditional Fuzzy . . . t-norms, t-conorms, etc. Zadeh’s Vision From Traditional . . . of Going from Fuzzy Zadeh’s Vision Zadeh’s Vision: . . . to Computing With Words: Home Page from the Idea’s Origin Title Page to Current Successes ◭◭ ◮◮ to Remaining Challenges ◭ ◮ Page 1 of 7 Vladik Kreinovich Go Back Department of Computer Science University of Texas at El Paso Full Screen El Paso, TX 79968, USA Close vladik@utep.edu Quit
1. The Origins of Fuzzy Techniques: Reminder The Origins of Fuzzy . . . Traditional Fuzzy . . . • Some experts are very skilled in medical diagnostics, t-norms, t-conorms, etc. control, etc. From Traditional . . . • Ideally: every patient should be diagnosed by the best Zadeh’s Vision doctor. Zadeh’s Vision: . . . • Problem: the best doctor does not have time to see all Home Page patients. Title Page • Solution: incorporate the expert knowledge into an au- ◭◭ ◮◮ tomatic system that everyone can use. ◭ ◮ • Problem: experts often describe their knowledge by us- Page 2 of 7 ing imprecise (“fuzzy”) words from natural language. Go Back • Examples: expert rules include conditions like “if a Full Screen tumor is small”, “if a car is far away and going fast”. Close Quit
2. Traditional Fuzzy Techniques (cont-d) The Origins of Fuzzy . . . Traditional Fuzzy . . . • Fuzzy logic is a technique for transforming imprecise t-norms, t-conorms, etc. expert rules into precise decision, precise control, etc. From Traditional . . . • Main idea: since we are not sure whether x is small, Zadeh’s Vision assign a degree of smallness to different values x . Zadeh’s Vision: . . . • In the computer: everything is represented as 0s and Home Page 1s; e.g., “true” is 1, “false” is 0. Title Page • We want degrees intermediate between 0 and 1, so it ◭◭ ◮◮ is natural to use numbers from [0 , 1]. ◭ ◮ • Elicitation: polling (probability-type), Likert scale, etc. Page 3 of 7 • Need to combine degrees: what is the degree to which Go Back a car is far away and going fast? Full Screen • Ideal solution: ask the expert about all possible com- binations of distance and speed. Close • Problem: there are too many combinations to ask about. Quit
3. t-norms, t-conorms, etc. The Origins of Fuzzy . . . Traditional Fuzzy . . . • Problem (reminder): we need to estimate degrees of t-norms, t-conorms, etc. A & B etc., and we cannot simply elicit them. From Traditional . . . • Solution: we estimate the degree of A & B based on Zadeh’s Vision degrees of A and B : d ( A & B ) = f & ( d ( A ) , d ( B )). Zadeh’s Vision: . . . • Details: requirements like A & B ≡ B & A and Home Page A & ( B & C ) ≡ ( A & B ) & C lead to t-norms. Title Page • Problem: there exist many different t-norms that sat- ◭◭ ◮◮ isfy all these requirements. ◭ ◮ • Details: different t-norms lead to different recommen- Page 4 of 7 dations. Go Back • t-norms are selected empirically (if selected at all :-), so that the elicited d ( A & B ) is the closest to f & ( d ( A ) , d ( B )). Full Screen • Example: medically best t-norm (MYCIN) turned out Close to be not appropriate for geophysics. Quit
4. From Traditional Fuzzy Logic to More Ade- The Origins of Fuzzy . . . quate Implementations of Computing With Words Traditional Fuzzy . . . t-norms, t-conorms, etc. • Problem: for the same statement, different experts pro- From Traditional . . . duce different degrees. Zadeh’s Vision • Traditional fuzzy logic: uses one of these degrees – or, Zadeh’s Vision: . . . e.g., their average. Home Page • Problem: an expert is not sure about his or her degree Title Page of belief in a statement: 71 or 72 on a scale 0–100? ◭◭ ◮◮ • Traditional fuzzy logic: if an expert selects between 7 ◭ ◮ and 8 on 1–10 scale, use 7.5. Page 5 of 7 • More adequate representation of expert uncertainty: Go Back – use range of possible values (interval-valued ap- Full Screen proach); – also indicate degrees to which different values from Close the range are possible (general type-2 approach). Quit
5. Zadeh’s Vision The Origins of Fuzzy . . . Traditional Fuzzy . . . • In many applications: the outcome is an imprecise con- t-norms, t-conorms, etc. clusion: e.g., the patient most probably has a flu. From Traditional . . . • How this is done now: Zadeh’s Vision Zadeh’s Vision: . . . – we start with words from natural language; – we transform them into numbers (intervals, etc.); Home Page – we process these numbers; and Title Page – we transform the resulting number into a natural ◭◭ ◮◮ language word describing the conclusion. ◭ ◮ • Why we use numbers: only because we know how to Page 6 of 7 process numbers. Go Back • Zadeh’s idea: cut the middleman: Full Screen – start with words, – process words, Close – produce the words as a result. Quit
6. Zadeh’s Vision: Challenges The Origins of Fuzzy . . . Traditional Fuzzy . . . • Ideally: we should operate directly with words. t-norms, t-conorms, etc. From Traditional . . . • Example: we should be able to add small and medium Zadeh’s Vision and get – what? Zadeh’s Vision: . . . • This is the gist of numerous Zadeh’s examples like Home Page – most Swedes are tall, Title Page – Johannes is a Swede, ◭◭ ◮◮ – what is the probability that Johannes is very tall? ◭ ◮ • Challenges: we are still far from this vision. Page 7 of 7 Go Back Full Screen Close Quit
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