1. 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.eduhttp://www-bisc.cs.berkeley.edu URL: http://www.cs.berkeley.edu/~zadeh/http://www.cs.berkeley.edu/~zadeh/ Email: Zadeh@cs.berkeley.eduZadeh@cs.berkeley.edu

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3. LAZ 2/22/2005 3/113 EVOLUTION OF COMPUTATION natural language arithmeticalgebra differential equations calculus differential equations numerical analysis symbolic computation computing with words precisiated natural language symbolic computation ++ ++ + + +

4. LAZ 2/22/2005 4/113 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?

5. LAZ 2/22/2005 5/113 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)

6. LAZ 2/22/2005 6/113 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)

7. LAZ 2/22/2005 7/113 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

8. LAZ 2/22/2005 8/113 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

9. LAZ 2/22/2005 9/113 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

10. LAZ 2/22/2005 10/113 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

11. LAZ 2/22/2005 11/113 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

12. LAZ 2/22/2005 12/113 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

13. LAZ 2/22/2005 13/113 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

14. LAZ 2/22/2005 14/113 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 q 1 : What is the number of white balls? q 2 : What is the probability that a ball drawn at random is white? q 1 and q 2 remain the same in the next version THE BALLS-IN-BOX PROBLEM

15. LAZ 2/22/2005 15/113 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 p 1 : there are about 20 balls p 2 : most are black p 3 : there are several times as many black balls as white balls PT’s solution?

16. LAZ 2/22/2005 16/113 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 q 1 : How many balls are large? q 2 : What is the probability that a ball drawn at random is large PT’s solution?

17. LAZ 2/22/2005 17/113 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: p 1 : there are about 20 balls p 2 : most are small p 3 : there are several times as many small balls as large balls q 1 : how many are large? q 2 : what is the probability that a ball drawn at random is large?

18. LAZ 2/22/2005 18/113 CONTINUED Version 5. Perception-based My perceptions are: p 1 : there are about 20 balls p 2 : most are large p 3 : if a ball is large then it is likely to be heavy q 1 : how many are heavy? q 2 : what is the probability that a ball drawn at random is not heavy?

19. LAZ 2/22/2005 19/113 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? 0.5 There is no underlying randomness 0.9 0.8 0.6 0.9 0.4

20. LAZ 2/22/2005 20/113 MEASUREMENT-BASED 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? PERCEPTION-BASED version 2

21. LAZ 2/22/2005 21/113 COMPUTATION (version 2) measurement-based X = number of black balls Y 2 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

22. LAZ 2/22/2005 22/113 FUZZY INTEGER PROGRAMMING x Y 1 X= several × y X= most × 20* X+Y= 20*

23. LAZ 2/22/2005 23/113 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 µ X (u) = µ >>a* (u) ^ µ <<b* (u)

24. LAZ 2/22/2005 24/113 PRECISIATION OF “approximately a,” *a x x x a a 2025 0 1 0 0 1 p  fuzzy graph probability distribution interval x 0 a possibility distribution  x a 0 1   s-precisiation singleton g-precisiation cg-precisiation

25. LAZ 2/22/2005 25/113 CONTINUED x p 0 bimodal distribution GCL-based (maximal generality) g-precisiation X isr R GC-form *a g-precisiation

26. LAZ 2/22/2005 26/113 VERA’S AGE PROBLEM q: How old is Vera? p 1 : Vera has a son, in mid-twenties p 2 : Vera has a daughter, in mid-thirties wk: the child-bearing age ranges from about 16 to about 42

27. LAZ 2/22/2005 27/113 CONTINUED *16 *41*42 *51 *67 *77 range 1 range 2 p1:p1: p2:p2: (p 1, p 2 ) 0 0 R(q/p 1, p 2, wk):  a =  ° *51  ° *67 timelines *a: approximately aHow is *a defined?

28. LAZ 2/22/2005 28/113 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, P 1 and P 2, P 1 is closer to the post office but it is in a yellow zone. If I park my car in P 1 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 P 2 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?

29. LAZ 2/22/2005 29/113 THE PARKING PROBLEM P0P0 P1P1 P2P2 P 1 : probability of arriving at the post office after it closes, starting in P 1 P t : probability of getting a ticket C t : cost of ticket P 2 : probability of arriving at the post office after it closes, starting in P 2 L: loss if package is not mailed

30. LAZ 2/22/2005 30/113 CONTINUED C t : expected cost of parking in P 1 C 1 = C t + p 1 L C 2 : expected cost of parking in C 2 C 2 = p 2 L standard approach: minimize expected cost standard approach is not applicable when the values of variables and parameters are perception-based (linguistic)

31. LAZ 2/22/2005 31/113 DEEP STRUCTURE (PROTOFORM) CtCt L L 0 P1P1 P2P2 Gain

32. LAZ 2/22/2005 32/113 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.

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34. LAZ 2/22/2005 34/113 BL PT PNL CW ++ bivalent logic probability theory precisiated natural language computing with words GTU CTP: computational theory of perceptions PFT: protoform theory PTp: perception-based probability theory THD: theory of hierarchical definability GTU: Generalized Theory of uncertainty CTP THD PFT PTp EXISTING TOOLS NEW TOOLS

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36. LAZ 2/22/2005 36/113 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.

37. LAZ 2/22/2005 37/113 BASIC FACET OF HUMAN COGNITION perceptions are intrinsically imprecise principal reasons: a) Bounded ability of sensory organs, and ultimately the brain, to resolve detail and store information b) Incompleteness of information perception *a (granule) approximately a perception of X X: attribute singleton a domain of X

38. LAZ 2/22/2005 38/113 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 INFORMATION measurement-based numerical perception-based linguistic measurement-based information may be viewed as a special case of perception-based information perception-based information is intrinsically imprecise

39. LAZ 2/22/2005 39/113 BASIC PERCEPTIONS / F-GRANULARITY 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 +… 1 0 size smallmedium large  a granule is a clump of attribute-values which are drawn together by indistinguishability, equivalence, similarity, proximity or functionality

40. LAZ 2/22/2005 40/113 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

41. LAZ 2/22/2005 41/113 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.

42. LAZ 2/22/2005 42/113 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

43. LAZ 2/22/2005 43/113 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

44. LAZ 2/22/2005 44/113 PERCEPTION OF RISK perception of function perception of likelihood perception of loss perception of risk Conventional definition of risk as the expected value of the loss function is an oversimplification

45. LAZ 2/22/2005 45/113 PERCEPTION OF MATHEMATICAL CONCEPTS: PERCEPTION OF FUNCTION 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 0 Y ff* : perception Y f* (fuzzy graph) medium x large f 0 SM L L M S granule

46. LAZ 2/22/2005 46/113 BIMODAL DISTRIBUTION (PERCEPTION-BASED PROBABILITY DISTRIBUTION) A1A1 A2A2 A3A3 P1P1 P2P2 P3P3 probability P(X) = P i(1) \A 1 + P i(2) \A 2 + P i(3) \A 3 Prob {X is A i } is P j(i) 0 X P(X)= low\small+high\medium+low\large

47. LAZ 2/22/2005 47/113 TEST PROBLEM A function, Y=f(X), is defined by its fuzzy graph expressed as f 1 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 0 SML L M S X Y M × L

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49. LAZ 2/22/2005 49/113 KEY MOTIVATION Basic objective Mechanization of reasoning Prerequisite Precisiation of meaning

50. LAZ 2/22/2005 50/113 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

51. LAZ 2/22/2005 51/113 KEY IDEA The point of departure in PNL is the key idea: A proposition, p, drawn from a natural language, NL, is precisiated by expressing its meaning as a generalized constraint In general, X, R, r are implicit in p precisiation of pexplicitation of X, R, r p X isr R constraining relation Identifier of modality (type of constraint) constrained (focal) variable

52. LAZ 2/22/2005 52/113 SIMPLE EXAMPLE Monika is young Age(Monika) is young Annotated representation X/Age(Monika) is R/young X R r (blank)

53. LAZ 2/22/2005 53/113 BASIC POINTS A proposition is an answer to a question example: p: Monika is young is an answer to the question q: How old is Monika? The concept of a generalized constraint serves as a basis for generalized-constraint- based semantics of natural languages

54. LAZ 2/22/2005 54/113 GENERALIZED CONSTRAINT (Zadeh 1986) Bivalent constraint (hard, inelastic, categorical:) X  C constraining bivalent relation X isr R constraining non-bivalent (fuzzy) relation index of modality (defines semantics) constrained variable Generalized constraint: r:  | = |  |  |  | … | blank | p | v | u | rs | fg | ps |… bivalentnon-bivalent (fuzzy)

55. LAZ 2/22/2005 55/113 CONTINUED constrained variable X is an n-ary variable, X= (X 1, …, X n ) X is a proposition, e.g., Leslie is tall X is a function of another variable: X=f(Y) X is conditioned on another variable, X/Y X has a structure, e.g., X= Location (Residence(Carol)) X is a generalized constraint, X: Y isr R X is a group variable. In this case, there is a group, G: (Name 1, …, Name n ), and each member of the group is associated with an attribute-value, A 1. X may be expressed symbolically as X: (Name 1 /A 1 +…+Name n /A n ) Basically, X is a relation

56. LAZ 2/22/2005 56/113 SIMPLE EXAMPLES “Check-out time is 1 pm,” is an instance of a generalized constraint on check-out time “Speed limit is 100km/h” is an instance of a generalized constraint on speed “Vera is a divorcee with two young children,” is an instance of a generalized constraint on Vera’s age

57. LAZ 2/22/2005 57/113 GENERALIZED CONSTRAINT—MODALITY r X isr R r: =equality constraint: X=R is abbreviation of X is=R r: ≤inequality constraint: X ≤ R r:  subsethood constraint: X  R r: blankpossibilistic constraint; X is R; R is the possibility distribution of X r: vveristic constraint; X isv R; R is the verity distribution of X r: pprobabilistic constraint; X isp R; R is the probability distribution of X

58. LAZ 2/22/2005 58/113 CONTINUED r: rsrandom set constraint; X isrs R; R is the set- valued probability distribution of X r: fgfuzzy graph constraint; X isfg R; X is a function and R is its fuzzy graph r: uusuality constraint; X isu R means usually (X is R) r: ggroup constraint; X isg R means that R constrains the attributed-values of the group

59. LAZ 2/22/2005 59/113 GENERALIZED CONSTRAINT—SEMANTICS A generalized constraint, GC, is associated with a test-score function, ts(u), which associates with each object, u, to which the constraint is applicable, the degree to which u satisfies the constraint. Usually, ts(u) is a point in the unit interval. However, if necessary, it may be an element of a semi-ring, a lattice, or more generally, a partially ordered set, or a bimodal distribution. example: possibilistic constraint, X is R X is RPoss(X=u) = µ R (u) ts(u) = µ R (u)

60. LAZ 2/22/2005 60/113 CONSTRAINT QUALIFICATION p isr R means r value of p is R in particular p isp R Prob(p) is R (probability qualification) p isv R Tr(p) is R (truth (verity) qualification) p is R Poss(p) is R (possibility qualification) examples (X is small) isp likely (X is small) is likely (X is small) isv very true ( X is small) is very true (X isu R) Prob(X is R) is usually

61. LAZ 2/22/2005 61/113 GENERALIZED CONSTRAINT LANGUAGE (GCL) GCL is an abstract language GCL is generated by combination, qualification and propagation of generalized constraints examples of elements of GCL (X isp R) and (X,Y) is S) (X isr R) is unlikely) and (X iss S) is likely If X is A then Y is B the language of fuzzy if-then rules is a sublanguage of GCL deduction= generalized constraint propagation

62. LAZ 2/22/2005 62/113 PRECISIATION = TRANSLATION INTO GCL annotation p X/A isr R/B GC-form of p example p: Carol lives in a small city near San Francisco X/Location(Residence(Carol)) is R/NEAR[City]  SMALL[City] pp* NLGCL precisiation translation GC-form GC(p)

63. LAZ 2/22/2005 63/113 GENERALIZED-CONSTRAINT-FORM(GC(p)) annotation p X/A isr R/B annotated GC(p) suppression X/A isr R/B X isr Risa deep structure (protoform) of p instantiation abstraction X isr R A isr B

64. LAZ 2/22/2005 64/113 PRECISIATION VIA TRANSLATION INTO GCL Examples: possibilistic Monika is young Age (Monika) is young Monika is much younger than Maria (Age (Monika), Age (Maria)) is much younger most Swedes are tall  Count (tall.Swedes/Swedes) is most XR X X R R

65. LAZ 2/22/2005 65/113 EXAMPLES: PROBABILISITIC X is a normally distributed random variable with mean m and variance  2 X isp N(m,  2 ) X is a random variable taking the values u 1, u 2, u 3 with probabilities p 1, p 2 and p 3, respectively X isp (p 1 \u 1 +p 2 \u 2 +p 3 \u 3 )

66. LAZ 2/22/2005 66/113 EXAMPLES: VERISTIC Robert is half German, quarter French and quarter Italian Ethnicity (Robert) isv (0.5|German + 0.25|French + 0.25|Italian) Robert resided in London from 1985 to 1990 Reside (Robert, London) isv [1985, 1990]

67. LAZ 2/22/2005 67/113 INFORMATION AND GENERALIZED CONSTRAINTS—KEY POINTS In CW, the carriers of information are propositions p: proposition GC(p): X isr R p is a carrier of information about X GC(p) is the information about X carried by p

68. LAZ 2/22/2005 68/113 BASIC STRUCTURE OF PNL p p*p** NLPFLGCL abstraction world knowledge module GCL: Generalized Constraint Language PFL: Protoform Language Dictionary 1: NL GCL Dictionary 2: GCL PFL Dictionary 3: NL to PFL deduction module summarization D2 D1 D3

69. LAZ 2/22/2005 69/113 WORLD KNOWLEDGE Examples icy roads are slippery big cars are safer than small cars usually it is hard to find parking near the campus on weekdays between 9 and 5 most Swedes are tall overeating causes obesity Ph.D. is the highest academic degree Princeton usually means Princeton University A person cannot have two fathers Netherlands has no mountains

70. LAZ 2/22/2005 70/113 WORLD KNOWLEDGE world knowledge—and especially knowledge about the underlying probabilities—plays an essential role in disambiguation, planning, search and decision processes what is not recognized to the extent that it should, is that world knowledge is for the most part perception-based KEY POINTS

71. LAZ 2/22/2005 71/113 WORLD KNOWLEDGE: EXAMPLES specific: if Robert works in Berkeley then it is likely that Robert lives in or near Berkeley if Robert lives in Berkeley then it is likely that Robert works in or near Berkeley generalized: if A/Person works in B/City then it is likely that A lives in or near B precisiated: Distance (Location (Residence (A/Person), Location (Work (A/Person) isu near protoform: F (A (B (C)), A (D (C))) isu R

72. LAZ 2/22/2005 72/113 THE CONCEPT OF A PROTOFORM AND ITS BASIC ROLE IN KNOWLEDGE REPRESENTATION, DEDUCTION AND SEARCH Informally, a protoform—abbreviation of prototypical form—is an abstracted summary. More specifically, a protoform is a symbolic expression which defines the deep semantic structure of a construct such as a concept, proposition, command, question, scenario, case or a system of such constructs Example: Monika is youngA(B) is C instantiation abstraction youngC

73. LAZ 2/22/2005 73/113 CONTINUED object p object space summarizationabstraction protoform protoform space summary of p S(p) A(S(p)) PF(p) PF(p): abstracted summary of p deep structure of p protoform equivalence protoform similarity

74. LAZ 2/22/2005 74/113 EXAMPLES Monika is youngAge(Monika) is young A(B) is C Monika is much younger than Robert (Age(Monika), Age(Robert) is much.younger D(A(B), A(C)) is E Usually Robert returns from work at about 6:15pm Prob{Time(Return(Robert)} is 6:15*} is usually Prob{A(B) is C} is D usually 6:15* Return(Robert) Time abstraction instantiation

75. LAZ 2/22/2005 75/113 PROTOFORMS at a given level of abstraction and summarization, objects p and q are PF-equivalent if PF(p)=PF(q) example p: Most Swedes are tallCount (A/B) is Q q: Few professors are richCount (A/B) is Q PF-equivalence class object space protoform space

76. LAZ 2/22/2005 76/113 EXAMPLES Alan has severe back pain. He goes to see a doctor. The doctor tells him that there are two options: (1) do nothing; and (2) do surgery. In the case of surgery, there are two possibilities: (a) surgery is successful, in which case Alan will be pain free; and (b) surgery is not successful, in which case Alan will be paralyzed from the neck down. Question: Should Alan elect surgery? Y X0 object Y X0 i-protoform option 1 option 2 0 1 2 gain

77. LAZ 2/22/2005 77/113 BASIC POINTS annotation: specification of class or type Monika is youngA(B) is C A/attribute of B, B/name, C/value of A abstraction has levels, just as summarization does most Swedes are tall most A’s are tall most A’s are B QA’s are B’s P and q are PF-equivalent (at level  ) iff they have identical protoforms (at level  ) most Swedes are tall = few professors are rich The concepts of cluster and mountain are PF- equivalent The concepts of risk and obesity are PF- equivalent

78. LAZ 2/22/2005 78/113 PF-EQUIVALENCE Scenario A: Alan has severe back pain. He goes to see a doctor. The doctor tells him that there are two options: (1) do nothing; and (2) do surgery. In the case of surgery, there are two possibilities: (a) surgery is successful, in which case Alan will be pain free; and (b) surgery is not successful, in which case Alan will be paralyzed from the neck down. Question: Should Alan elect surgery?

79. LAZ 2/22/2005 79/113 PF-EQUIVALENCE Scenario B: Alan needs to fly from San Francisco to St. Louis and has to get there as soon as possible. One option is fly to St. Louis via Chicago and the other through Denver. The flight via Denver is scheduled to arrive in St. Louis at time a. The flight via Chicago is scheduled to arrive in St. Louis at time b, with a b. Question: Which option is best?

80. LAZ 2/22/2005 80/113 THE TRIP-PLANNING PROBLEM I have to fly from A to D, and would like to get there as soon as possible I have two choices: (a) fly to D with a connection in B; or (b) fly to D with a connection in C if I choose (a), I will arrive in D at time t 1 if I choose (b), I will arrive in D at time t 2 t 1 is earlier than t 2 therefore, I should choose (a) ? A C B D (a) (b)

81. LAZ 2/22/2005 81/113 PROTOFORM EQUIVALENCE options gain c 12 a b 0

82. LAZ 2/22/2005 82/113 PROTOFORM EQUIVALENCE Backpain= trip planning= divorce= job change

83. LAZ 2/22/2005 83/113 PROTOFORMAL SEARCH RULES example query: What is the distance between the largest city in Spain and the largest city in Portugal? protoform of query: ?Attr (Desc(A), Desc(B)) procedure query: ?Name (A)|Desc (A) query: Name (B)|Desc (B) query: ?Attr (Name (A), Name (B))

84. LAZ 2/22/2005 84/113 PNL AS A DEFINITION / DESCRIPTION / SPECIFICATION LANGUAGE X: concept, description, specification Describe X in a natural language Precisiate description of X KEY IDEA Test: What is the definition of a mountain?

85. LAZ 2/22/2005 85/113 FUZZY CONCEPTS Relevance Causality Summary Cluster Mountain Valley In the existing literature, there are no operational definitions of these concepts

86. LAZ 2/22/2005 86/113 DIGRESSION: COINTENSION C human perception of C p(C) definition of C d(C) intension of p(C)intension of d(C) CONCEPT cointension: coincidence of intensions of p(C) and d(C)

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88. LAZ 2/22/2005 88/113 PROTOFORMAL DEDUCTION pqpq p* q* NLGCL precisiation p** q** PFL summarization precisiationabstraction answer a r** World Knowledge Module WKM DM deduction module

89. LAZ 2/22/2005 89/113 protoformal rule symbolic partcomputational part FORMAT OF PROTOFORMAL DEDUCTION RULES

90. LAZ 2/22/2005 90/113 Rules of deduction in the Deduction Database (DDB) are protoformal examples: (a) compositional rule of inference PROTOFORMAL DEDUCTION X is A (X, Y) is B Y is A ° B symbolic computational (b) extension principle X is A Y = f(X) Y = f(A) symbolic Subject to: computational

91. LAZ 2/22/2005 91/113 PROTOFORM DEDUCTION RULE: GENERALIZED MODUS PONENS X is A If X is B then Y is C Y is D D = A ° (B×C) D = A ° (B  C) computational 1 computational 2 symbolic (fuzzy graph; Mamdani) (implication; conditional relation) classical A AB B fuzzy logic

92. LAZ 2/22/2005 92/113 X is A (X, Y) is B Y is A  B symbolic part computational part subject to: Prob (X is A) is B Prob (X is C) is D PROTOFORMAL RULES OF DEDUCTION examples

93. LAZ 2/22/2005 93/113 COUNT-AND MEASURE-RELATED RULES Q A’s are B’s ant (Q) A’s are not B’s r 0 1 1 ant (Q) Q Q A’s are B’s Q 1/2 A’s are 2 B’s r 0 1 1 Q Q 1/2 most Swedes are tall ave (height) Swedes is ?h Q A’s are B’s ave (B|A) is ?C, crisp  

94. LAZ 2/22/2005 94/113 CONTINUED not(QA’s are B’s) (not Q) A’s are B’s Q 1 A’s are B’s Q 2 (A&B)’s are C’s Q 1 Q 2 A’s are (B&C)’s Q 1 A’s are B’s Q 2 A’s are C’s (Q 1 +Q 2 -1) A’s are (B&C)’s

95. LAZ 2/22/2005 95/113 PROTOFORMAL CONSTRAINT PROPAGATION Dana is young Age (Dana) is young X is A p GC(p)PF(p) Tandy is a few years older than Dana Age (Tandy) is (Age (Dana)) Y is (X+B) X is A Y is (X+B) Y is A+B Age (Tandy) is (young+few) +few

96. LAZ 2/22/2005 96/113 THE TALL SWEDES PROBLEM p: Most Swedes are tall q: What is the average height of Swedes PNL-based solution 1. PF(p): Count(A/B) is Q PF(q): H(A) is C no match in Deduction Database excessive summarization 2. PF(p):  Count(P[H is A] / P) is Q H ave is C

97. LAZ 2/22/2005 97/113 CONTINUED NameH Name 1h 1.. Name Nh n P: Name Hµ a Name i h i µ A (h i ) P[H is A]:, subject to:

98. LAZ 2/22/2005 98/113 Rules of deduction are basically rules governing generalized constraint propagation The principal rule of deduction is the extension principle RULES OF DEDUCTION X is A f(X,) is B Subject to: computationalsymbolic

99. LAZ 2/22/2005 99/113 GENERALIZATIONS OF THE EXTENSION PRINCIPLE f(X) is A g(X) is B Subject to: information = constraint on a variable given information about X inferred information about X

100. LAZ 2/22/2005 100/113 CONTINUED f(X 1, …, X n ) is A g(X 1, …, X n ) is B Subject to: (X 1, …, X n ) is A g j (X 1, …, X n ) is Y j, j=1, …, n (Y 1, …, Y n ) is B Subject to:

101. LAZ 2/22/2005 101/113 MODULAR DEDUCTION DATABASE POSSIBILITY MODULE PROBABILITY MODULE SEARCH MODULE FUZZY LOGIC MODULE agent FUZZY ARITHMETIC MODULE EXTENSION PRINCIPLE MODULE

102. LAZ 2/22/2005 102/113 SUMMATION humans have a remarkable capability—a capability which machines do not have—to perform a wide variety of physical and mental tasks using only perceptions, with no measurements and no computations perceptions are intrinsically imprecise, reflecting the bounded ability of sensory organs, and ultimately the brain, to resolve detail and store information KEY POINTS

103. LAZ 2/22/2005 103/113 CONTINUED imprecision of perceptions stands in the way of constructing a computational theory of perceptions within the conceptual structure of bivalent logic and bivalent-logic-based probability theory this is why existing scientific theories— based as they are on bivalent logic and bivalent-logic-based probability theory— provide no tools for dealing with perception- based information

104. LAZ 2/22/2005 104/113 CONTINUED in computing with words (CW), the objects of computation are propositions drawn from a natural language and, in particular, propositions which are descriptors of perceptions computing with words is a methodology which may be viewed as (a) a new direction for dealing with imprecision, uncertainty and partial truth; and (b) as a basis for the analysis and design of systems which are capable of operating on perception-based information

105. LAZ 2/22/2005 105/113

106. LAZ 2/22/2005 106/113 January 26, 2005 Factual Information About the Impact of Fuzzy Logic PATENTS  Number of fuzzy-logic-related patents applied for in Japan: 17,740  Number of fuzzy-logic-related patents issued in Japan: 4,801  Number of fuzzy-logic-related patents issued in the US: around 1,700

107. LAZ 2/22/2005 107/113 PUBLICATIONS Count of papers containing the word “fuzzy” in title, as cited in INSPEC and MATH.SCI.NET databases. Compiled by Camille Wanat, Head, Engineering Library, UC Berkeley, December 22, 2004 Number of papers in INSPEC and MathSciNet which have "fuzzy" in their titles: INSPEC - "fuzzy" in the title 1970-1979: 569 1980-1989: 2,404 1990-1999: 23,207 2000-present: 14,172 Total: 40,352 MathSciNet - "fuzzy" in the title 1970-1979: 443 1980-1989: 2,465 1990-1999: 5,483 2000-present: 3,960 Total: 12,351

108. LAZ 2/22/2005 108/113 JOURNALS (“fuzzy” or “soft computing” in title) 1. Fuzzy Sets and Systems 2. IEEE Transactions on Fuzzy Systems 3. Fuzzy Optimization and Decision Making 4. Journal of Intelligent & Fuzzy Systems 5. Fuzzy Economic Review 6. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 7. Journal of Japan Society for Fuzzy Theory and Systems 8. International Journal of Fuzzy Systems 9. Soft Computing 10. International Journal of Approximate Reasoning--Soft Computing in Recognition and Search 11. Intelligent Automation and Soft Computing 12. Journal of Multiple-Valued Logic and Soft Computing 13. Mathware and Soft Computing 14. Biomedical Soft Computing and Human Sciences 15. Applied Soft Computing

109. LAZ 2/22/2005 109/113 APPLICATIONS The range of application-areas of fuzzy logic is too wide for exhaustive listing. Following is a partial list of existing application-areas in which there is a record of substantial activity. 1.Industrial control 2.Quality control 3.Elevator control and scheduling 4.Train control 5.Traffic control 6.Loading crane control 7.Reactor control 8.Automobile transmissions 9.Automobile climate control 10.Automobile body painting control 11.Automobile engine control 12.Paper manufacturing 13.Steel manufacturing 14.Power distribution control 15.Software engineerinf 16.Expert systems 17.Operation research 18.Decision analysis 19.Financial engineering 20.Assessment of credit-worthiness 21.Fraud detection 22.Mine detection 23.Pattern classification 24.Oil exploration 25.Geology 26.Civil Engineering 27.Chemistry 28.Mathematics 29.Medicine 30.Biomedical instrumentation 31.Health-care products 32.Economics 33.Social Sciences 34.Internet 35.Library and Information Science

110. LAZ 2/22/2005 110/113 Product Information Addendum 1 This addendum relates to information about products which employ fuzzy logic singly or in combination. The information which is presented came from SIEMENS and OMRON. It is fragmentary and far from complete. Such addenda will be sent to the Group from time to time. SIEMENS: * washing machines, 2 million units sold * fuzzy guidance for navigation systems (Opel, Porsche) * OCS: Occupant Classification System (to determine, if a place in a car is occupied by a person or something else; to control the airbag as well as the intensity of the airbag). Here FL is used in the product as well as in the design process (optimization of parameters). * fuzzy automobile transmission (Porsche, Peugeot, Hyundai) OMRON: * fuzzy logic blood pressure meter, 7.4 million units sold, approximate retail value $740 million dollars Note: If you have any information about products and or manufacturing which may be of relevance please communicate it to Dr. Vesa Niskanen vesa.a.niskanen@helsinki.fi and Masoud Nikravesh Nikravesh@cs.berkeley.edu.vesa.a.niskanen@helsinki.fiNikravesh@cs.berkeley.edu

111. LAZ 2/22/2005 111/113 Product Information Addendum 2 This addendum relates to information about products which employ fuzzy logic singly or in combination. The information which is presented came from Professor Hideyuki Takagi, Kyushu University, Fukuoka, Japan. Professor Takagi is the co-inventor of neurofuzzy systems. Such addenda will be sent to the Group from time to time. Facts on FL-based systems in Japan (as of 2/06/2004) 1. Sony's FL camcorders Total amount of camcorder production of all companies in 1995-1998 times Sony's market share is the following. Fuzzy logic is used in all Sony's camcorders at least in these four years, i.e. total production of Sony's FL-based camcorders is 2.4 millions products in these four years. 1,228K units X 49% in 1995 1,315K units X 52% in 1996 1,381K units X 50% in 1997 1,416K units X 51% in 1998 2. FL control at Idemitsu oil factories Fuzzy logic control is running at more than 10 places at 4 oil factories of Idemitsu Kosan Co. Ltd including not only pure FL control but also the combination of FL and conventional control. They estimate that the effect of their FL control is more than 200 million YEN per year and it saves more than 4,000 hours per year.

112. LAZ 2/22/2005 112/113 3. Canon Canon used (uses) FL in their cameras, camcorders, copy machine, and stepper alignment equipment for semiconductor production. But, they have a rule not to announce their production and sales data to public. Canon holds 31 and 31 established FL patents in Japan and US, respectively. 4. Minolta cameras Minolta has a rule not to announce their production and sales data to public, too. whose name in US market was Maxxum 7xi. It used six FL systems in a camera and was put on the market in 1991 with 98,000 YEN (body price without lenses). It was produced 30,000 per month in 1991. Its sister cameras, alpha-9xi, alpha-5xi, and their successors used FL systems, too. But, total number of production is confidential.

113. LAZ 2/22/2005 113/113 5. FL plant controllers of Yamatake Corporation Yamatake-Honeywell (Yamatake's former name) put FUZZICS, fuzzy software package for plant operation, on the market in 1992. It has been used at the plants of oil, oil chemical, chemical, pulp, and other industries where it is hard for conventional PID controllers to describe the plan process for these more than 10 years. They planed to sell the FUZZICS 20 - 30 per year and total 200 million YEN. As this software runs on Yamatake's own control systems, the software package itself is not expensive comparative to the hardware control systems. 6. Others Names of 225 FL systems and products picked up from news articles in 1987 - 1996 are listed at http://www.adwin.com/elec/fuzzy/note_10.html in Japanese.) http://www.adwin.com/elec/fuzzy/note_10.html Note: If you have any information about products and or manufacturing which may be of relevance please communicate it to Dr. Vesa Niskanen vesa.a.niskanen@helsinki.fi and Masoud Nikravesh Nikravesh@cs.berkeley.edu, with cc to me.vesa.a.niskanen@helsinki.fi Nikravesh@cs.berkeley.edu