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Computing with Words and its Applications to Information Processing, Decision and Control

Computing with Words and its Applications to Information Processing, Decision and Control Lotfi A. Zadeh Computer Science Division Department of EECS UC Berkeley February 28, 2005 University of Vienna URL: http://www-bisc.cs.berkeley.edu URL: http://www.cs.berkeley.edu/~zadeh/

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Computing with Words and its Applications to Information Processing, Decision and Control

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  1. Computing with Words and its Applications to Information Processing, Decision and Control Lotfi A. Zadeh Computer Science Division Department of EECSUC Berkeley February 28, 2005 University of Vienna URL: http://www-bisc.cs.berkeley.edu URL: http://www.cs.berkeley.edu/~zadeh/ Email: Zadeh@cs.berkeley.edu

  2. BACKDROP LAZ 2/22/2005

  3. EVOLUTION OF COMPUTATION natural language arithmetic algebra + + algebra calculus differential equations + + differential equations numerical analysis symbolic computation + + symbolic computation computing with words precisiated natural language + LAZ 2/22/2005

  4. COMPUTING WITH WORDS (CW) • In computing with words (CW), the objects of computation are words and propositions drawn from a natural language example: X and Y are real-valued variables and f is a function from R to R which is described in words: f: if X is small then Y is small if X is medium then Y is large if X is large then Y is small What is the maximum of f? LAZ 2/22/2005

  5. COMPUTING WITH WORDS (CW) • The centerpiece of CW is the concept of a generalized constraint (Zadeh 1986) • In CW, the traditional view that information is statistical in nature is put aside. Instead, a much more general view of information is adopted: information is a generalized constraint, with statistical information constituting a special case • The point of departure in CW is representation of the meaning of a proposition drawn from a natural language as a a generalized constraint • CW is based on fuzzy logic (FL) LAZ 2/22/2005

  6. FROM BIVALENT LOGIC, BL, TO FUZZY LOGIC, FL • In classical, Aristotelian, bivalent logic, BL, truth is bivalent, implying that every proposition, p, is either true or false, with no degrees of truth allowed • In multivalent logic, ML, truth is a matter of degree • In fuzzy logic, FL: • everything is, or is allowed to be, a matter of degree • everything is, or is allowed to be imprecise (approximate) • everything is, or is allowed to be, granular (linguistic) LAZ 2/22/2005

  7. NUMBERS ARE RESPECTED—WORDS ARE NOT • It is a deep-seated tradition in science to equate scientific progress to progression from perceptions to measurements and from the use of words to the use of numbers. • Computing with words is a challenge to this tradition • Computing with words opens the door to a wide-ranging enlargement of the role of natural languages in scientific theories—in particular, in decision analysis, medicine and economics LAZ 2/22/2005

  8. IN QUEST OF PRECISION • Reducing smog would save lives, Bay report says (San Francisco Examiner) • Expected to attract national attention, the Santa Clara Criteria Air Pollutant Benefit Analysis is the first to quantify the effects on health of air pollution in California • Removing lead from gasoline could save the lives of 26.7 Santa Clara County residents and spare them 18 strokes, 27 heart attacks, 722 nervous system problems and 1,668 cases where red blood cell production is affected LAZ 2/22/2005

  9. CONTINUED • Study projects S.F. 5-year AIDS toll (S.F. Chronicle July 15, 1992) • The report projects that the number of new AIDS cases will reach a record 2,173 this year and decline thereafter to 2,007 new cases in 1997 LAZ 2/22/2005

  10. QUEST FOR PRECISION Qualitative Analysis for Management H. Bender et al • There is a 60% chance that the survey results will be positive Prob(success) = 0.600 • Throughout the text, probabilities and utilities are treated as exact numbers LAZ 2/22/2005

  11. COMPUTING WITH WORDS LEVEL 1: Computing with Words • most × many • small + large LEVEL 2: Computing with Propositions • Most Swedes are tall • It is unlikely to rain in San Francisco in midsummer LEVEL 2: Employs generalized-constraint-based semantics of natural languages LAZ 2/22/2005

  12. KEY POINTS • Computing with words is not a replacement for computing with numbers; it is an addition • Use of computing with words is a necessity when the available information is perception-based or not precise enough to justify the use of numbers • Use of computing with words is advantageous when there is a tolerance for imprecision which can be exploited to achieve tractability, simplicity, robustness and reduced cost LAZ 2/22/2005

  13. BASIC POINTS • In computing with words, the objects of computation are words, propositions, and perceptions described in a natural language • A natural language is a system for describing perceptions • In CW, a perception is equated to its description in a natural language LAZ 2/22/2005

  14. THE BALLS-IN-BOX PROBLEM Version 1. Measurement-based A flat box contains a layer of black and white balls. You can see the balls and are allowed as much time as you need to count them • q1: What is the number of white balls? • q2: What is the probability that a ball drawn at random is white? • q1 and q2 remain the same in the next version LAZ 2/22/2005

  15. CONTINUED Version 2. Perception-based You are allowed n seconds to look at the box. n seconds is not enough to allow you to count the balls You describe your perceptions in a natural language p1: there are about 20 balls p2: most are black p3: there are several times as many black balls as white balls PT’s solution? LAZ 2/22/2005

  16. CONTINUED Version 3. Measurement-based The balls have the same color but different sizes You are allowed as much time as you need to count the balls q1: How many balls are large? q2: What is the probability that a ball drawn at random is large PT’s solution? LAZ 2/22/2005

  17. CONTINUED Version 4. Perception-based You are allowed n seconds to look at the box. n seconds is not enough to allow you to count the balls Your perceptions are: p1: there are about 20 balls p2: most are small p3: there are several times as many small balls as large balls q1: how many are large? q2: what is the probability that a ball drawn at random is large? LAZ 2/22/2005

  18. CONTINUED Version 5. Perception-based My perceptions are: p1: there are about 20 balls p2: most are large p3: if a ball is large then it is likely to be heavy q1: how many are heavy? q2: what is the probability that a ball drawn at random is not heavy? LAZ 2/22/2005

  19. A SERIOUS LIMITATION OF PT • Version 4 points to a serious short coming of PT • In PT there is no concept of cardinality of a fuzzy set • How many large balls are in the box? 0.6 0.8 0.4 0.9 0.9 0.5 • There is no underlying randomness LAZ 2/22/2005

  20. a box contains 20 black and white balls over seventy percent are black there are three times as many black balls as white balls what is the number of white balls? what is the probability that a ball picked at random is white? a box contains about 20 black and white balls most are black there are several times as many black balls as white balls what is the number of white balls what is the probability that a ball drawn at random is white? MEASUREMENT-BASED PERCEPTION-BASED version 2 LAZ 2/22/2005

  21. measurement-based X = number of black balls Y2 number of white balls X  0.7 • 20 = 14 X + Y = 20 X = 3Y X = 15 ; Y = 5 p =5/20 = .25 perception-based X = number of black balls Y = number of white balls X = most × 20* X = several *Y X + Y = 20* P = Y/N COMPUTATION (version 2) LAZ 2/22/2005

  22. FUZZY INTEGER PROGRAMMING Y X= most × 20* X+Y= 20* X= several × y x 1 LAZ 2/22/2005

  23. PARTIAL EXISTENCE • X, a and b are real numbers, with a  b • Find an X or X’s such that X  a* and X  b* • a*: approximately a; b*: approximately b • Fuzzy logic solution • Partial existence is not a probabilistic concept µX(u) = µ>>a*(u)^ µ<<b* (u) LAZ 2/22/2005

  24. PRECISIATION OF “approximately a,” *a  1 singleton s-precisiation 0 a x  1 cg-precisiation interval 0 a x p probability distribution 0 g-precisiation a x  possibility distribution 0 a x  1 fuzzy graph 0 20 25 LAZ 2/22/2005 x

  25. CONTINUED p bimodal distribution g-precisiation GCL-based (maximal generality) 0 x g-precisiation *a X isr R GC-form LAZ 2/22/2005

  26. VERA’S AGE PROBLEM • q: How old is Vera? • p1: Vera has a son, in mid-twenties • p2: Vera has a daughter, in mid-thirties • wk: the child-bearing age ranges from about 16 to about 42 LAZ 2/22/2005

  27. CONTINUED range 1 timelines p1: 0 *16 *41 *42 *67 range 2 p2: 0 *16 *42 *51 *77 (p1, p2) *16 *42 *51 *67 R(q/p1, p2, wk): a=  °*51  °*67 *a: approximately a How is *a defined? LAZ 2/22/2005

  28. THE PARKING PROBLEM • I have to drive to the post office to mail a package. The post office closes at 5 pm. As I approach the post office, I come across two parking spots, P1 and P2, P1 is closer to the post office but it is in a yellow zone. If I park my car in P1 and walk to the post office, I may get a ticket, but it is likely that I will get to the post office before it closes. If I park my car in P2 and walk to the post office, it is likely that I will not get there before the post office closes. Where should I park my car? LAZ 2/22/2005

  29. THE PARKING PROBLEM P0 P1 P2 P1: probability of arriving at the post office after it closes, starting in P1 Pt: probability of getting a ticket Ct: cost of ticket P2 : probability of arriving at the post office after it closes, starting in P2 L: loss if package is not mailed LAZ 2/22/2005

  30. CONTINUED • Ct: expected cost of parking in P1 • C1 = Ct + p1L • C2 : expected cost of parking in C2 • C2 = p2L • standard approach: minimize expected cost • standard approach is not applicable when the values of variables and parameters are perception-based (linguistic) LAZ 2/22/2005

  31. DEEP STRUCTURE (PROTOFORM) Gain P1 P2 0 Ct L L LAZ 2/22/2005

  32. THE NEED FOR NEW TOOLS • The balls-in-box problem and the parking problem are simple examples of problems which do not lend themselves to analysis by conventional techniques based on bivalent logic and probability theory. The principal source of difficulty is that, more often than not, decision-relevant information is a mixture of measurements and perceptions. Perception-based information is intrinsically imprecise, reflecting the bounded ability of human sensory organs, and ultimately the brain, to resolve detail and store information. To deal with perceptions, new tools are needed. The principal tool is the methodology of computing with words (CW). The centerpiece of new tools is the concept of a generalized constraint. LAZ 2/22/2005

  33. New Tools LAZ 2/22/2005

  34. NEW TOOLS EXISTING TOOLS computing with words bivalent logic + + CW BL PNL PT precisiated natural language probability theory CTP PFT GTU THD CTP: computational theory of perceptions PFT: protoform theory PTp: perception-based probability theory THD: theory of hierarchical definability GTU: Generalized Theory of uncertainty PTp LAZ 2/22/2005

  35. COMPUTING WITH PERCEPTIONS LAZ 2/22/2005

  36. PERCEPTIONS • Perceptions play a key role in human cognition. Humans—but not machines—have a remarkable capability to perform a wide variety of physical and mental tasks without any measurements and any computations. Everyday examples of such tasks are driving a car in city traffic, playing tennis and summarizing a book. LAZ 2/22/2005

  37. BASIC FACET OF HUMAN COGNITION X: attribute perception of X • perceptions are intrinsically imprecise • principal reasons: • Bounded ability of sensory organs, and ultimately the brain, to resolve detail and store information • Incompleteness of information perception singleton a *a (granule) approximately a domain of X LAZ 2/22/2005

  38. INFORMATION measurement-based numerical perception-based linguistic • it is 35 C° • Over 70% of Swedes are taller than 175 cm • probability is 0.8 • It is very warm • Most Swedes are tall • probability is high • it is cloudy • traffic is heavy • it is hard to find parking near the campus • measurement-based information may be viewed as a special case of perception-based information • perception-based information is intrinsically imprecise LAZ 2/22/2005

  39. BASIC PERCEPTIONS / F-GRANULARITY • a granule is a clump of attribute-values which are drawn together by indistinguishability, equivalence, similarity, proximity or functionality • temperature: warm+cold+very warm+much warmer+… • time: soon + about one hour + not much later +… • distance: near + far + much farther +… • speed: fast + slow +much faster +… • length: long + short + very long +…  small medium large 1 LAZ 2/22/2005 0 size

  40. CONTINUED • similarity: low + medium + high +… • possibility: low + medium + high + almost impossible +… • likelihood: likely + unlikely + very likely +… • truth (compatibility): true + quite true + very untrue +… • count: many + few + most + about 5 (5*) +… subjective probability = perception of likelihood LAZ 2/22/2005

  41. COMPUTING WITH PERCEPTIONS • One of the major aims of CW is to serve as a basis for equipping machines with a capability to operate on perception-based information. A key idea in CW is that of dealing with perceptions through their descriptions in a natural language. In this way, computing and reasoning with perceptions is reduced to operating on propositions drawn from a natural language. LAZ 2/22/2005

  42. BASIC PERCEPTIONS attributes of physical objects • distance • time • speed • direction • length • width • area • volume • weight • height • size • temperature sensations and emotions • color • smell • pain • hunger • thirst • cold • joy • anger • fear concepts • count • similarity • cluster • causality • relevance • risk • truth • likelihood • possibility LAZ 2/22/2005

  43. DEEP STRUCTURE OF PERCEPTIONS • perception of likelihood • perception of truth (compatibility) • perception of possibility (ease of attainment or realization) • perception of similarity • perception of count (absolute or relative) • perception of causality • perception of risk subjective probability = quantified perception of likelihood LAZ 2/22/2005

  44. PERCEPTION OF RISK perception of function perception of likelihood perception of risk perception of loss • Conventional definition of risk as the expected • value of the loss function is an oversimplification LAZ 2/22/2005

  45. PERCEPTION OF MATHEMATICAL CONCEPTS: PERCEPTION OF FUNCTION granule L M S 0 S M L Y f 0 Y medium x large f* (fuzzy graph) perception f f* : if X is small then Y is small if X is medium then Y is large if X is large then Y is small 0 X LAZ 2/22/2005

  46. BIMODAL DISTRIBUTION (PERCEPTION-BASED PROBABILITY DISTRIBUTION) probability P3 P2 P1 X 0 A2 A1 A3 P(X) = Pi(1)\A1 +Pi(2)\A2 + Pi(3)\A3 Prob {X is Ai } is Pj(i) P(X)= low\small+high\medium+low\large LAZ 2/22/2005

  47. TEST PROBLEM • A function, Y=f(X), is defined by its fuzzy graph expressed as f1 if X is small then Y is small if X is medium then Y is large if X is large then Y is small (a) what is the value of Y if X is not large? (b) what is the maximum value of Y Y M × L L M S X 0 LAZ 2/22/2005 S M L

  48. PNL LAZ 2/22/2005

  49. KEY MOTIVATION • Basic objective • Mechanization of reasoning • Prerequisite • Precisiation of meaning LAZ 2/22/2005

  50. PRECISIATION OF MEANING • Use with adequate ventilation • Speed limit is 100km/hr • Most Swedes are tall • Take a few steps • Monika is young • Beyond reasonable doubt • Overeating causes obesity • Relevance • Causality • Mountain • Most • Usually LAZ 2/22/2005

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