Equations Describing . . . Need to Describe . . . Fuzzy Logic as a . . . High Concentrations Chemical Kinetics and . . . Case of High . . . Naturally Lead to How to Observe the . . . 3rd Derivative in . . . Fuzzy-Type Interactions and What Type of . . . to Gravitational Wave Bursts Home Page Title Page Oscar Galindo, Olga Kosheleva, and Vladik Kreinovich ◭◭ ◮◮ University of Texas at El Paso, El Paso, Texas 79968, USA, ◭ ◮ ogalindomo@miners.utep.edu, olgak@utep.edu, vladik@utep.edu Page 1 of 39 Go Back Full Screen Close Quit
Equations Describing . . . 1. Equations Describing the Physical World Need to Describe . . . Fuzzy Logic as a . . . • Traditionally, our knowledge is described in precise Chemical Kinetics and . . . terms, from Newton’s laws to relativity theory. Case of High . . . • In many physical situations, it is not possible to exactly How to Observe the . . . predict the future state of a system. 3rd Derivative in . . . What Type of . . . • In some situations: Home Page – we know the exact equations describing the inter- Title Page action of particles, ◭◭ ◮◮ – but the number of particles is so huge that it is not possible to exactly solve this system of equations. ◭ ◮ • For example, a room full of air contains about 10 23 Page 2 of 39 molecules. Go Back • In this case, all we can do is make predictions about Full Screen the frequency of different outcomes. Close Quit
Equations Describing . . . 2. Physical Equations (cont-d) Need to Describe . . . Fuzzy Logic as a . . . • In other words, we can only make predictions about Chemical Kinetics and . . . the probabilities of different events. Case of High . . . • This is the case of statistical physics . How to Observe the . . . 3rd Derivative in . . . • If we take quantum effects into account, then the same What Type of . . . phenomenon occurs for all possible physical processes. Home Page • Indeed, according to quantum physics: Title Page – it is not even theoretically possible ◭◭ ◮◮ – to predict the exact future values of all the physical ◭ ◮ quantities such as coordinates, momentum, etc. Page 3 of 39 • All we can do is predict the corresponding probabili- Go Back ties. Full Screen Close Quit
Equations Describing . . . 3. Physical Equations Describing the Probabili- Need to Describe . . . ties Are Also Exact Fuzzy Logic as a . . . Chemical Kinetics and . . . • While the knowledge is probabilistic, equations describ- Case of High . . . ing how these probabilities change are exact. How to Observe the . . . • This is true for Boltzmann’s equations of statistical 3rd Derivative in . . . physics and for Schroedinger’s quantum equations. What Type of . . . Home Page • These equations are usually smooth (differentiable). Title Page • Different particles are reasonable independent. ◭◭ ◮◮ • So the overall probability can be obtained by multiply- ◭ ◮ ing probabilities corresponding to different particles. Page 4 of 39 • The product function is differentiable infinitely many times. Go Back • So usually, the corresponding equations are also differ- Full Screen entiable (smooth). Close Quit
Equations Describing . . . 4. Need to Describe Expert Knowledge Need to Describe . . . Fuzzy Logic as a . . . • In many real-life situations: Chemical Kinetics and . . . – in addition to (or, sometimes, instead of) the exact Case of High . . . equations, How to Observe the . . . – we also have imprecise (“fuzzy”) expert knowledge, 3rd Derivative in . . . What Type of . . . – knowledge that experts describe by using imprecise Home Page natural-language words. Title Page • For example, a medical doctor may say that a skin irritation is suspicious if it has irregular shape. ◭◭ ◮◮ ◭ ◮ • A medicine is recommended if the patient has a high fever. Page 5 of 39 • However, what exactly is irregular or high is not well- Go Back defined, it is subjective. Full Screen Close Quit
Equations Describing . . . 5. Fuzzy Logic as a Natural Way to Describe Im- Need to Describe . . . precise Expert Knowledge Fuzzy Logic as a . . . Chemical Kinetics and . . . • In both above examples, it is not the case that we have Case of High . . . an exact threshold on temperature, so that: How to Observe the . . . – below this threshold, we have one decision, and 3rd Derivative in . . . – above the threshold, we have another decision. What Type of . . . Home Page • This would make no sense: Title Page – why give a medicine to someone whose body tem- ◭◭ ◮◮ perature is 39.00 C ◭ ◮ – but not to someone whose temperature is 38.99 C? Page 6 of 39 • For temperatures close to some transition value: Go Back – experts are not 100% sure whether the temperature is high (or whether the shape is irregular), Full Screen – they are only confident to some degree . Close Quit
Equations Describing . . . 6. Fuzzy Logic (cont-d) Need to Describe . . . Fuzzy Logic as a . . . • In the computer, “absolutely true” is usually repre- Chemical Kinetics and . . . sented as 1, and “absolutely false” as 0. Case of High . . . • So it is reasonable to describe intermediate degrees of How to Observe the . . . confidence by intermediate numbers, from [0 , 1]. 3rd Derivative in . . . • This is the main idea behind fuzzy logic . What Type of . . . Home Page • This formalism was invented by Lotfi A. Zadeh to de- Title Page scribe imprecise expert knowledge. ◭◭ ◮◮ • In fuzzy logic, to describe each imprecise property P like “high”, we assign, ◭ ◮ – to each possible value q of the corresponding quan- Page 7 of 39 tity, Go Back – a number µ ( q ) from the interval [0 , 1] that describes Full Screen to what extent the expert is confident in P ( q ), Close – e.g., to what extent the given temperature q is high. Quit
Equations Describing . . . 7. Fuzzy Degrees Need to Describe . . . Fuzzy Logic as a . . . • How can we estimate the expert’s degrees of confi- Chemical Kinetics and . . . dence? Case of High . . . • We can, e.g., ask each expert to mark his/her degree How to Observe the . . . of confidence on a scale from 0 to 10: 3rd Derivative in . . . What Type of . . . – 0 meaning no confidence at all, and Home Page – 10 meaning absolutely sure. Title Page • To get a value between 0 and 1, we divide the resulting ◭◭ ◮◮ estimate by 10. ◭ ◮ • Experts can estimate degrees of certainty in their state- ments. Page 8 of 39 • However, conclusions based on expert knowledge often Go Back take into account several expert statements. Full Screen Close Quit
Equations Describing . . . 8. Fuzzy Degrees (cont-d) Need to Describe . . . Fuzzy Logic as a . . . • Our degree of confidence in such a conclusion is thus Chemical Kinetics and . . . equal to our degree of confidence that: Case of High . . . – the first of used statements is true and How to Observe the . . . – the second used statement is true, etc. 3rd Derivative in . . . What Type of . . . • In other words, Home Page – in addition to the expert’s degrees of confidence in Title Page their statements S 1 , . . . , S n , ◭◭ ◮◮ – we also need to estimate the degrees of confidence in “and”-combinations S i & S j , S i & S j & S k , etc. ◭ ◮ • In the ideal world, we can ask the experts to estimate Page 9 of 39 the degree of confidence in each such combination. Go Back • However, this is not realistically possible. Full Screen • Indeed, for n original statements, there are 2 n − 1 such Close combinations. Quit
Equations Describing . . . 9. Fuzzy Degrees (cont-d) Need to Describe . . . Fuzzy Logic as a . . . • Indeed, combinations are in 1-1 correspondence with Chemical Kinetics and . . . non-empty subsets of the set of n statements. Case of High . . . • Already for reasonable n = 30, we get an astronomical How to Observe the . . . number 2 30 ≈ 10 9 combinations. 3rd Derivative in . . . What Type of . . . • There is no way that we can ask a billion questions to Home Page the experts. Title Page • We cannot elicit the expert’s degree of confidence in “and”-combinations directly from the experts. ◭◭ ◮◮ ◭ ◮ • So, we need to estimate these degrees based on the experts’ degrees of confidence in each statement S i . Page 10 of 39 Go Back Full Screen Close Quit
Equations Describing . . . 10. Fuzzy Degrees (cont-d) Need to Describe . . . Fuzzy Logic as a . . . • In other words, we need to be able: Chemical Kinetics and . . . – to combine the degrees of confidence a and b of Case of High . . . statements A and B How to Observe the . . . – into an estimate for degree of confidence in the 3rd Derivative in . . . “and”-combination A & B . What Type of . . . Home Page • The algorithm for such combination is called an “and”- Title Page operation or, for historical reasons, a t-norm . ◭◭ ◮◮ • The result of applying this combination algorithm to numbers a and b will be denoted f & ( a, b ). ◭ ◮ Page 11 of 39 Go Back Full Screen Close Quit
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