Studying Physical World Newton’s Physics: . . . Resulting Model This Model Leads to . . . Formalizing the Informal, From Equations to . . . Precisiating the Imprecise: Divergence: A Problem How Fuzzy Logic Can Help Maybe Fuzzy . . . Mathematicians and Physicists Fuzzy and Physics: . . . by Formalizing Their Future Is Fuzzy! Home Page Intuitive Ideas Title Page Vladik Kreinovich ◭◭ ◮◮ Department of Computer Science ◭ ◮ University of Texas at El Paso 500 W. University Page 1 of 33 El Paso, TX 79968, USA Go Back vladik@utep.edu Full Screen talk based on joint work with Olga Kosheleva and Renata Reiser Close Quit
Studying Physical World Newton’s Physics: . . . 1. Outline Resulting Model • Fuzzy methodology: This Model Leads to . . . From Equations to . . . – transforms expert ideas – formulated in terms of Divergence: A Problem words from natural language, Maybe Fuzzy . . . – into precise rules and formulas. Fuzzy and Physics: . . . • In this talk, we show that by applying this methodol- Future Is Fuzzy! ogy to intuitive physical and mathematical ideas: Home Page – we can get known fundamental physical equations Title Page and ◭◭ ◮◮ – we can get known mathematical techniques for solv- ◭ ◮ ing these equations. Page 2 of 33 • This makes us confident that in the future, fuzzy tech- Go Back niques will still help physicists and mathematicians. Full Screen Close Quit
Studying Physical World Newton’s Physics: . . . 2. Fuzzy Is Most Successful When We Have Par- Resulting Model tial Knowledge This Model Leads to . . . • Fuzzy methodology has been invented to transform: From Equations to . . . Divergence: A Problem – expert ideas – formulated in terms of words from Maybe Fuzzy . . . natural language, Fuzzy and Physics: . . . – into precise rules and formulas, rules and formulas Future Is Fuzzy! understandable by a computer. Home Page • Fuzzy methodology has led to many successful appli- Title Page cations, especially in intelligent control. ◭◭ ◮◮ • Major successes of fuzzy methodology is when we only ◭ ◮ have partial knowledge. Page 3 of 33 • This is true for all known fuzzy control success stories: Go Back washing machines, camcoders, elevators, trains, etc. Full Screen Close Quit
Studying Physical World Newton’s Physics: . . . 3. Is Fuzzy Poor Man Data Processing? Resulting Model • From this viewpoint: This Model Leads to . . . From Equations to . . . – as we gain more knowledge about a system, Divergence: A Problem – a moment comes when we do not need to use fuzzy Maybe Fuzzy . . . techniques any longer. Fuzzy and Physics: . . . – we will be able to use traditional (crisp) techniques. Future Is Fuzzy! Home Page • So, fuzzy techniques look like a (successful but still) intermediate step, Title Page – “poor man’s” data processing techniques, ◭◭ ◮◮ – that need to be used only if we cannot apply “more ◭ ◮ optimal” traditional methods. Page 4 of 33 • We show, on example of the study of physical world, Go Back that fuzzy methodology can be very useful beyond that. Full Screen Close Quit
Studying Physical World Newton’s Physics: . . . 4. Studying Physical World Resulting Model • When we study the physical world, our first task is This Model Leads to . . . physical : to find the physical laws . From Equations to . . . Divergence: A Problem • In precise terms, these are equations that describe how Maybe Fuzzy . . . the values of physical quantities change with time. Fuzzy and Physics: . . . • Once we have found these equations, the next task is Future Is Fuzzy! mathematical : Home Page – we need to solve these equations Title Page – to predict the future values of physical quantities. ◭◭ ◮◮ • Both tasks are not easy. In both tasks, we: ◭ ◮ – start with informal ideas, and Page 5 of 33 – gradually move to exact equations and exact algo- Go Back rithms for solving these equations. Full Screen • But such precisiation of informal ideas is exactly what fuzzy techniques were invented for, so let’s use them. Close Quit
Studying Physical World Newton’s Physics: . . . 5. Newton’s Physics: Informal Description Resulting Model • A body usually tries to go to the points x where its This Model Leads to . . . potential energy V ( x ) is the smallest. From Equations to . . . Divergence: A Problem • For example, a moving rock on the mountain tries to Maybe Fuzzy . . . go down. Fuzzy and Physics: . . . • The sum of the potential energy V ( x ) and the kinetic Future Is Fuzzy! energy K is preserved: Home Page 3 � 2 K = 1 � dx i � 2 · m · . Title Page dt i =1 ◭◭ ◮◮ • Thus, when the body minimizes its potential energy, it ◭ ◮ thus tries to maximize its kinetic energy. Page 6 of 33 • We will show that when we apply the fuzzy techniques Go Back to this informal description, we get Newton’s equations m · d 2 x i dt 2 = − ∂V Full Screen . ∂x i Close Quit
Studying Physical World Newton’s Physics: . . . 6. First Step: Selecting a Membership Function Resulting Model • The body tries to get to the areas where the potential This Model Leads to . . . energy V ( x ) is small. From Equations to . . . Divergence: A Problem • We need to select the corresponding membership func- Maybe Fuzzy . . . tion µ ( V ). Fuzzy and Physics: . . . • For example, we can poll several ( n ) experts and if Future Is Fuzzy! n ( V ) of them consider V small, take µ ( V ) = n ( V ) . Home Page n Title Page • In physics, we only know relative potential energy – relative to some level. ◭◭ ◮◮ • If we change that level by V 0 , we replace V by V + V 0 . ◭ ◮ • So, values V and V + V 0 represent the same value of Page 7 of 33 the potential energy – but for different levels. Go Back • A seemingly natural formalization: µ ( V ) = µ ( V + V 0 ). Full Screen • Problem: we get useless µ ( V ) = const. Close Quit
Studying Physical World Newton’s Physics: . . . 7. Re-Analyzing the Polling Method Resulting Model • In the poll, the more people we ask, the more accurate This Model Leads to . . . is the resulting opinion. From Equations to . . . Divergence: A Problem • Thus, to improve the accuracy of the poll, we add m Maybe Fuzzy . . . folks to the original n top experts. Fuzzy and Physics: . . . • These m extra folks may be too intimidated by the Future Is Fuzzy! original experts. Home Page • With the new experts mute, we still have the same Title Page number n ( V ) of experts who say “yes”. ◭◭ ◮◮ • As a result, instead of the original value µ ( V ) = n ( V ) , ◭ ◮ n we get µ ′ ( V ) = n ( V ) n Page 8 of 33 n + m = c · µ ( V ), where c = n + m. Go Back • These two membership functions µ ( V ) and µ ′ ( V ) = Full Screen c · µ ( V ) represent the same expert opinion. Close Quit
Studying Physical World Newton’s Physics: . . . 8. Resulting Formalization of the Physical Intu- Resulting Model ition This Model Leads to . . . • How to describe that potential energy is small? From Equations to . . . Divergence: A Problem • Idea: value V and V + V 0 are equivalent – they differ Maybe Fuzzy . . . by a starting level for measuring potential energy. Fuzzy and Physics: . . . • Conclusion: membership functions µ ( V ) and µ ( V + V 0 ) Future Is Fuzzy! should be equivalent. Home Page • We know: membership functions µ ( V ) and µ ′ ( V ) are Title Page equivalent if µ ′ ( V ) = c · µ ( V ). ◭◭ ◮◮ • Hence: for every V 0 , there is a value c ( V 0 ) for which ◭ ◮ µ ( V + V 0 ) = c ( V 0 ) · µ ( V ) . Page 9 of 33 • It is known that the only monotonic solution to this Go Back equation is µ ( V ) = a · exp( − k · V ) . Full Screen • So we will use this membership function to describe that the potential energy is small. Close Quit
Studying Physical World Newton’s Physics: . . . 9. Resulting Formalization of the Physical Intu- Resulting Model ition (cont-d) This Model Leads to . . . • Reminder: we use µ ( V ) = a · exp( − k · V ) to describe From Equations to . . . that potential energy is small. Divergence: A Problem Maybe Fuzzy . . . • How to describe that kinetic energy is large? Fuzzy and Physics: . . . • Idea: K is large if − K is small. Future Is Fuzzy! • Resulting membership function: Home Page µ ( K ) = exp( − k · ( − K )) = exp( k · K ) . Title Page ◭◭ ◮◮ • We want to describe the intuition that ◭ ◮ – the potential energy is small and Page 10 of 33 – that the kinetic energy is large and – that the same is true at different moments of time. Go Back Full Screen • According to fuzzy methodology, we must therefore se- lect an appropriate “and”-operation (t-norm) f & ( a, b ). Close Quit
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