Fuzzy Logic: Reminder From the . . . From the Intuitive . . . Fuzzy Data . . . Why Complex-Valued When Are Integration . . . Relation to Complex . . . Fuzzy? Why Complex Other Reasons Why . . . Values in General? Why Complex . . . Complex Numbers . . . A Computational Home Page Explanation Title Page ◭◭ ◮◮ Olga Kosheleva and Vladik Kreinovich ◭ ◮ University of Texas at El Paso, El Paso, TX 79968, USA olgak@utep.edu, vladik@utep.edu Page 1 of 14 Thavatchai Ngamsantivong Go Back King Mongkut’s Univ. of Technology North Bangkok, Thailand Full Screen tvc@kmutnb.ac.th Close Quit
Fuzzy Logic: Reminder From the . . . 1. Fuzzy Logic: Reminder From the Intuitive . . . • In classical (2-valued) logic, every statement is either Fuzzy Data . . . true or false. When Are Integration . . . Relation to Complex . . . • In the computer, “true” is usually represented as 1, Other Reasons Why . . . and “false” as 0. Why Complex . . . • The resulting 2-valued logic { 0 , 1 } is well equipped to Complex Numbers . . . represent: Home Page – situations when we are completely confident that a Title Page given statement is true, and ◭◭ ◮◮ – situations when we are completely confident that a ◭ ◮ given statement is false. Page 2 of 14 • 2-valued logic cannot adequately represent situations when we only have some degree of confidence. Go Back • To describe such situations, L. Zadeh invented fuzzy Full Screen logic. Close Quit
Fuzzy Logic: Reminder From the . . . 2. From the Mathematical Viewpoint, Complex- From the Intuitive . . . Valued Fuzzy Sets Are Natural Fuzzy Data . . . • In the original version of fuzzy logic, the set of possible When Are Integration . . . truth values is an interval [0 , 1]. Relation to Complex . . . Other Reasons Why . . . • [0 , 1]-based logic assumes that we can describe an ex- Why Complex . . . pert’s degree of confidence by an exact number. Complex Numbers . . . • Real-life experts often cannot meaningfully distinguish Home Page between nearby numbers. Title Page • We thus need to generalize the set of all real numbers ◭◭ ◮◮ from the interval [0 , 1]. ◭ ◮ • In mathematics, one of the natural generalizations of Page 3 of 14 real numbers are complex numbers. Go Back • Not surprisingly, complex-valued generalizations of fuzzy sets have been proposed and successfully used. Full Screen Close Quit
Fuzzy Logic: Reminder From the . . . 3. From the Intuitive Viewpoint, Complex-Valued From the Intuitive . . . Fuzzy Sets Remain a Puzzle Fuzzy Data . . . • Fuzzy sets are not just a mathematical theory. When Are Integration . . . Relation to Complex . . . • They are an intuitively clear way to describe how we Other Reasons Why . . . humans deal with uncertainty. Why Complex . . . • The original idea is very natural: describe possible ex- Complex Numbers . . . pert’s degrees of confidence ranging all the way: Home Page – from “absolutely false” (0) Title Page – to “absolutely true” (1). ◭◭ ◮◮ • In contrast, the idea of using complex numbers is not ◭ ◮ clear at all. Page 4 of 14 • Why complex-valued fuzzy numbers are useful is thus Go Back still largely a mystery. Full Screen Close Quit
Fuzzy Logic: Reminder From the . . . 4. Fuzzy Data Processing: Computational Chal- From the Intuitive . . . lenges Fuzzy Data . . . • The most widely used fuzzy-related techniques are tech- When Are Integration . . . niques of fuzzy control and fuzzy modeling. Relation to Complex . . . Other Reasons Why . . . • In these techniques, we start with rules, then: Why Complex . . . – we use t-norm and t-conorm to combine member- Complex Numbers . . . ship functions corresponding to these rules, and Home Page – we apply defuzzification to the resulting member- Title Page ship function to get a control value u . ◭◭ ◮◮ • For defuzzification, we can choose u for which µ ( u ) → � u · µ ( u ) du ◭ ◮ max or take u = µ ( u ) du . � Page 5 of 14 • Combining membership functions can be done in par- Go Back allel and thus, really fast. Full Screen • Integration and global optimization are NP-hard, so Close they are the main challenges in fuzzy data processing. Quit
Fuzzy Logic: Reminder From the . . . 5. When Are Integration and Maximization Prob- From the Intuitive . . . lems Feasible Fuzzy Data . . . • NP-hard means that we cannot solve all particular When Are Integration . . . cases of the integration and maximization problem. Relation to Complex . . . Other Reasons Why . . . • A natural question is: for what classes of functions can we feasibly solve these problems? Why Complex . . . Complex Numbers . . . • These problems are known to be feasible for triangular Home Page functions and not feasible for general ones. Title Page • Where is the “threshold” separating feasible from non- ◭◭ ◮◮ feasible cases? ◭ ◮ • Such a threshold have discovered in a recent paper by Kawamura et al.: Page 6 of 14 – both integration and optimization problems are NP- Go Back hard for smooth (differentiable) functions; Full Screen – these two problems become feasible (= polynomial- Close time) for analytical functions. Quit
Fuzzy Logic: Reminder From the . . . 6. Relation to Complex Numbers From the Intuitive . . . • For functions of a real variable, analytical functions are Fuzzy Data . . . power series: f ( x ) = a 0 + a 1 · ( x − x 0 )+ a 2 · ( x − x 0 ) 2 + . . . When Are Integration . . . Relation to Complex . . . • All such functions (exp( x ), sin( x ), etc.) can be natu- Other Reasons Why . . . rally extended to complex numbers. Why Complex . . . • In the complex domain, analytical functions can be Complex Numbers . . . defined as functions f ( z ) differentiable w.r.t. z . Home Page • So, a function f ( x ) is analytical if it can be extended Title Page to a smooth functions of a complex variable. ◭◭ ◮◮ • Thus, the above threshold result can be formulated as ◭ ◮ follows: Page 7 of 14 – for general smooth functions of a real variable, both Go Back integration and optimization problems are NP-hard; – for f-s extendable to smooth functions of a complex Full Screen variable, integration and optimization are feasible. Close Quit
Fuzzy Logic: Reminder From the . . . 7. Resulting Computational Explanation of Why From the Intuitive . . . Complex-Valued Fuzzy Fuzzy Data . . . • We thus arrive at the computational explanation of When Are Integration . . . why complex-valued fuzzy sets are practically useful: Relation to Complex . . . Other Reasons Why . . . • The most computationally intense operations involved Why Complex . . . in fuzzy techniques are: Complex Numbers . . . – integration and Home Page – optimization. Title Page • These operations are computationally feasible only when: ◭◭ ◮◮ – the corresponding functions ◭ ◮ – can be extended to smooth functions of complex Page 8 of 14 variables. Go Back Full Screen Close Quit
Fuzzy Logic: Reminder From the . . . 8. Other Reasons Why Complex Numbers Are From the Intuitive . . . Useful: They Are Easier to Use Fuzzy Data . . . • Many real-life phenomena – from planets orbiting the When Are Integration . . . Sun to waves – are periodic in time. Relation to Complex . . . Other Reasons Why . . . • A periodic signal x ( t ) can be expanded into Fourier Why Complex . . . series: Complex Numbers . . . ∞ � x ( t ) = x 0 + ( s n · sin( n · ω · t ) + c n · cos( n · ω · t )) . Home Page n =1 Title Page • This same linear combination can be equivalently rep- ◭◭ ◮◮ resented in complex-valued form as ◭ ◮ � ∞ � � Page 9 of 14 x ( t ) = Re z n · exp(i · n · ω · t ) . j =0 Go Back • Here, e.g., shifting t → t + t 0 is easy for complex num- Full Screen bers, somewhat more complex for sin( x ) and cos( x ). Close Quit
Fuzzy Logic: Reminder From the . . . 9. Why Complex Numbers in General? From the Intuitive . . . • Complex numbers are ubiquitous in engineering and in Fuzzy Data . . . physics (quantum physics, control); why? When Are Integration . . . Relation to Complex . . . • Physical theories are described in terms of differential Other Reasons Why . . . equations . Why Complex . . . • To solve the corr. physical problems, we need to solve Complex Numbers . . . ( integrate ) the corresponding differential equations. Home Page • Alternatively, physical theories can be describing in Title Page terms of optimization principles . ◭◭ ◮◮ • Control also often means optimization. ◭ ◮ • Integration and optimization become feasible only if Page 10 of 14 the functions are extendable to complex domain. Go Back • Here is our answer to the “why?” question: the use of Full Screen complex numbers guarantees that we have feasibility. Close Quit
Fuzzy Logic: Reminder From the . . . 10. Why Complex Numbers: An Answer That We From the Intuitive . . . Gave So Far Fuzzy Data . . . • We explained the effectiveness of complex numbers by When Are Integration . . . citing a result about integration and optimization: Relation to Complex . . . Other Reasons Why . . . – in the general case, the problem is NP-hard, so no Why Complex . . . feasible algorithm is possible; Complex Numbers . . . – on the other, when functions can be extended to the Home Page complex domain, feasible algorithms are possible. Title Page ◭◭ ◮◮ ◭ ◮ Page 11 of 14 Go Back Full Screen Close Quit
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