1. Precisiation of Meaning—From Natural Language to Granular Computing Lotfi A. Zadeh Computer Science Division Department of EECS UC Berkeley August 14, 2010 GrC’10 San Jose, CA Abridged version URL: http://www.cs.berkeley.edu/~zadeh/http://www.cs.berkeley.edu/~zadeh/ Email: Zadeh@eecs.berkeley.eduZadeh@eecs.berkeley.edu Research supported in part by ONR Grant N00014-02-1-0294, BT Grant CT1080028046, Omron Grant, Tekes Grant, Azerbaijan Ministry of Communications and Information Technology Grant, Azerbaijan University of Azerbaijan Republic and the BISC Program of UC Berkeley. 1 /152LAZ 8/6/2010

2. 2 /152LAZ 8/6/2010

3. PREAMBLE How does precisiation of meaning relate to granular computing? At first glance, there is no connection. But let us take a closer look. The coming decade will be a decade of automation of everyday reasoning and decision-making. In a world of automated reasoning and decision- making, computation with natural language is certain to play a prominent role. 3 /152LAZ 8/6/2010

4. CONTINUED Computation with natural language is closely related to Computing with Words (CW or CWW). Basically, CW is a system of computation which offers and important capability which traditional systems of computation do not have—the capability to compute with information described in a natural language. 4 /152LAZ 8/6/2010

5. CONTINUED Much of human knowledge is described in natural language. Basically, a natural language is a system for describing perceptions. Perceptions are intrinsically imprecise, reflecting the bounded ability of human sensory organs, and ultimately the brain, to resolve detail and store information. Imprecision of perceptions is passed on to natural languages. 5 /152LAZ 8/6/2010

6. CONTINUED Imprecision of natural languages is a major obstacle to computation with natural language. Raw (unprecisiated) natural language cannot be computed with. A prerequisite to computation with natural language is precisiation of meaning. 6 /152LAZ 8/6/2010

7. CW= [PRECISIATION COMPUTATION] computationprecisiation CW q I Ans(q/I) q* I* precisiation module computation module Phase 1Phase 2 fuzzy logic Precisiation and computation employ the machinery of fuzzy logic. However, the machinery which is employed is not that of traditional fuzzy logic. Granular computing 7 /152LAZ 8/6/2010

8. CONTINUED What is employed in CW is a new version of fuzzy logic. The cornerstones of new fuzzy logic are graduation, granulation, precisiation and computation. FUZZY LOGIC graduationgranulation precisiation computation 8 /152LAZ 8/6/2010

9. A BRIEF EXPOSITION OF NEW FUZZY LOGIC The point of departure in fuzzy logic is the concept of a fuzzy set. Informally, a fuzzy set is a class with unsharp boundary. class set generalization fuzzy set 9 /152LAZ 8/6/2010

10. fuzziness = unsharpness of class boundaries In the real world, fuzziness is a pervasive phenomenon. To construct better models of reality it’s necessary to develop a better understanding of how to deal precisely with unsharpness of class boundaries. In large measure, fuzzy logic is motivated by this need. KEY POINTS 10 /152LAZ 8/6/2010

11. A fuzzy set, A, in a space, U, is precisiated through graduation, that is, association with a membership function, μ A, which assigns to each object, u, in U its grade of membership, μ A (u), in U. Usually μ A (u) is a number in the [0,1], in which case A is a fuzzy set of type 1. A is a fuzzy set of type 2 if μ A (u) is a fuzzy set of type 1. CONTINUED 11 /152LAZ 8/6/2010

12. THE CONCEPT OF GRADUATION Graduation of a fuzzy concept or a fuzzy set, A, serves as a means of precisiation of A. Examples Graduation of middle-age Graduation of the concept of earthquake via the Richter Scale Graduation of recession? Graduation of mountain? 12 /152LAZ 8/6/2010

13. EXAMPLE—MIDDLE-AGE Imprecision of meaning = elasticity of meaning Elasticity of meaning = fuzziness of meaning 40 6045 55 μ 1 0 definitely middle-age definitely not middle-age 43 0.8 middle-age core of middle-age 13 /152LAZ 8/6/2010

14. GRADUATION graduation declarativeexperiential verificationelicitation Human-Machine Communication (HMC) Human-Human Communication (HHC) 14 /152LAZ 8/6/2010

15. HMC—HONDA FUZZY LOGIC TRANSMISSION Control Rules: 1.If (speed is low) and (shift is high) then (-3) 2.If (speed is high) and (shift is low) then (+3) 3.If (throt is low) and (speed is high) then (+3) 4.If (throt is low) and (speed is low) then (+1) 0 1 Speed ThrottleShift 30130 Grade 180 0 1 Grade 54 0 1 Grade 5 Close Low Fuzzy Set High Low Not Low Not Very Low 15 /152LAZ 8/6/2010

16. GRADUATION—ELICITATION Humans have a remarkable capability to graduate perceptions, that is, to associate perceptions with degrees on a scale. It is this capability that is exploited for elicitation of membership functions. Example Robert tells me that Vera is middle-aged. What does Robert mean by middle-aged? More specifically, what is the membership function that Robert associates with middle- aged? I elicit the membership function from Robert by asking a series of questions. 16 /152LAZ 8/6/2010

17. GRADUATION—ELICITATION Procedure Typical question: What is the degree to which a particular age, say 43, fits your perception of middle-aged? Please mark the degree on a scale from 0 to 1 using a Z-mouse. 17 /152LAZ 8/6/2010

18. Z-MOUSE—A VISUAL MEANS OF ENTRY AND RETRIEVAL OF FUZZY DATA A Z-mouse is an electronic implementation of a spray pen. The cursor is a round fuzzy mark called an f-mark. The color of the mark is a matter of choice. A dot identifies the centroid of the mark. The cross-section of a f-mark is a trapezoidal fuzzy set with adjustable parameters. imprecise probability gain optimism/ pessimism preference 0 0 0 0 1 *0.8 *65 *.8 *0.5 *0.8 0 risk - aversion I Ans(q/I) 1 1 1 f-mark 18 /152LAZ 8/6/2010

19. THE CONCEPT OF GRANULATION The concept of granulation is unique to fuzzy logic and plays a pivotal role in its applications. The concept of granulation is inspired by the way in which humans deal with imprecision, uncertainty and complexity. Granulation serves as a means of imprecisiation (coarsening of information). 19 /152LAZ 8/6/2010

20. GRADUATION / GRANULATION A A* *A graduation/precisiation granulation/imprecisiation graduation = precisiation granulation = imprecisiation 20 /152LAZ 8/6/2010

21. BASIC CONCEPTS—GRANULE Informally, a granule in a universe of discourse, U, is a clump of elements of U drawn together by indistinguishability, equivalence, similarity, proximity or functionality. A granule is precisiated through association with a generalized constraint. *A granule U universe of discourse 21 /152LAZ 8/6/2010

22. BASIC CONCEPTS—SINGULAR AND GRANULAR VALUES 7.3%high.8high 160/80high singular granular unemployment probability blood pressure *A granular value of X singular value of X universe of discourse U A 22 /152LAZ 8/6/2010

23. BASIC CONCEPTS—SINGULAR AND GRANULAR VARIABLES A singular variable, X, is a variable which takes values in U, that is, the values of X are singletons in U. A granular variable, *X, is a variable whose values are granules in U. A linguistic variable, *X, is a granular variable with linguistic labels for granular values. A quantized variable is a special case of a granular variable. 23 /152LAZ 8/6/2010

24. EXAMPLE Age as a singular variable takes values in the interval [0,120]. *Age as a granular (linguistic) variable takes as values fuzzy subsets of [0,120] labeled young, middle-aged, old, not very young, etc. quantizedAge 0 1 µ 1 0 young middle -aged old Age µ granulated 24 /152LAZ 8/6/2010

25. GRANULATION—KEY POINTS Granulation is closely related to coarsening of information, and to summarization. Granulation is a transformation which may be applied to any object, A A*A Three closely related meanings of granulation. (a) Granulation applied to a singular value (singular to granular transformation). granulation *A granular value of X singular value of X universe of discourse U a 25 /152LAZ 8/6/2010

26. SINGULATION (GRANULAR TO SINGULAR TRANSFORMATION) Singulation is inverse of granulation. Centroidal singulation A p p A 26 /152LAZ 8/6/2010

27. CONTINUED (b) Granulation applied to a singular variable, X, transforms X into a granular variable, *X. X *X (c) Granulation of a fuzzy set granulation A granule partition covering *A granule 27 /152LAZ 8/6/2010

28. (d) GRANULATION OF A FUNCTION GRANULATION=SUMMARIZATION 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 f*f : perception Y *f (fuzzy graph) medium × large f 0 SM L L M S granule summarization X 28 /152LAZ 8/6/2010

29. (e) GRANULAR VS. GRANULE-VALUED DISTRIBUTIONS distribution … p1p1 granules pnpn P1P1 P2P2 P PnPn P 0 A1A1 A2A2 AAnAn X g(u): probability density of X possibility distribution of probability distributions probability distribution of possibility distributions 29 /152LAZ 8/6/2010

30. MODES OF GRANULATION granulation forceddeliberate Forced: singular values of variables are not known. Deliberate: singular values of variables are known. There is a tolerance for imprecision. Precision carries a cost. Granular values are employed to reduce cost. 30 /152LAZ 8/6/2010

31. FUZZY LOGIC GAMBIT Fuzzy Logic Gambit = deliberate granulation followed by graduation The Fuzzy Logic Gambit is employed in most of the applications of fuzzy logic in the realm of consumer products 0 Y f X if X is small then Y is small if X is medium then Y is large if X is large then Y is small granulation summarization 31 /152LAZ 8/6/2010

32. 32 /152LAZ 8/6/2010

33. PRECISIATION OF PROPOSITIONS The concept of a proposition is one of the most basic concepts in the realms of both natural and synthetic languages. A dictionary definition of a proposition reads: an expression in language or signs of something that can be believed, doubted, denied or is either true or false. In CW, this definition is put aside. 33 /152LAZ 8/6/2010

34. CONTINUED Familiar examples: Robert is very bright Leslie is much taller than Ixel Most Swedes are tall. 34 /152LAZ 8/6/2010

35. CONTINUED As was noted earlier, precisiation of meaning is a prerequisite to computation with information described in a natural language. The issue of precisiation of propositions has a position of centrality in CW. Precisiation of propositions is beyond the reach of traditional approaches to semantics of natural languages. 35 /152LAZ 8/6/2010

36. CONTINUED In CW, precisiation of propositions is carried out through the use of what is referred to as generalized-constraint- based semantics, GCS. GCS is not needed for Level 1 CW. In Level 1 CW, the objects of computation are simple propositions, in particular, propositions which involve linguistic variable and fuzzy if-then rules. Level 2 CW is concerned with general propositions drawn from a natural language. 36 /152LAZ 8/6/2010

37. LEVEL 1 CW VS LEVEL 2 CW Example Level 1 CW: If X is small and Y is medium then (X+Y) is (small+medium) Level 2 CW: If most Swedes are tall and most tall Swedes are blond then most×most Swedes are blond. 37 /152LAZ 8/6/2010

38. A CW-BASED APPROACH TO PRECISIATION OF PROPOSITIONS— KEY POINTS Let p be a proposition drawn from a natural language. p is a carrier of information. Information is a restriction (constraint) on the values which a variable can take. In application to p, three basic questions arise. 38 /152LAZ 8/6/2010

39. CONTINUED 1. What is the constrained variable in p? Call it X. 2. What is the constraining relation in p? Call it R. 3. How does R constrain X? Call it r. In GCS, a proposition is represented as: p X isr R X isr R is a generalized constraint 39 /152LAZ 8/6/2010

40. CONTINUED Examples: X  2  5 X is a standard constraint. Standard constraints have no elasticity. Usually X is larger than approximately 2 and smaller than approximately 5, is a generalized constraint. Generalized constraints have elasticity. Elasticity is needed for representation of meaning of propositions drawn from a natural language. 40 /152LAZ 8/6/2010

41. KEY POINT The concept of a generalized constraint bridges the divide between linguistics and mathematics. 41 /152LAZ 8/6/2010

42. COINTENSIVE PRECISIATION Informally, cointension is a qualitative measure of proximity of the meanings of p and p*, with the object of precisiation, p, and the result of precisiation, p*, referred to as the precisiend and precisiand, respectively. 42 /152LAZ 8/6/2010

43. precisiendprecisiationprecisiand BASIC CONCEPTS IN PRECISIATION precisiation language system p: object of precisiation precisiand = model of meaning of precisiend precisiation ≈ modelization intension = attribute-based meaning cointension = measure of proximity of meanings = measure of proximity of the model and the object of modelization p*: result of precisiation cointension 43 /152LAZ 8/6/2010

44. A precisiend has a multiplicity of precisiands. Generally, achievement of cointensive precisiation requires that if the precisiend is fuzzy so must be the precisiand. Crisp definitions of fuzzy concepts is the norm in science. What is widely unrecognized is that generally crisp definitions of fuzzy concepts are not cointensive. COINTENSION PRINCIPLE 44 /152LAZ 8/6/2010

45. SUMMARY A key idea which underlies precisiation of meaning in CW—an idea which differentiates precisiation of meaning in CW from traditional approaches to representation of meaning in natural languages—is that of representing the computational model of p as a generalized constraint. 45 /152LAZ 8/6/2010

46. GENERALIZED CONSTRAINT= COMPUTATIONAL MODEL OF p pX isr R R is a restriction on the values of X. Typically, R is a fuzzy granule. r defines the way in which R constrains X. constrained variable copulaconstraining relation precisiation 46 /152LAZ 8/6/2010

47. PRIMARY GENERALIZED CONSTRAINTS In most applications, especially in the realm of natural languages, only three primary constraints and their combinations are employed. Primary constraints: possibilistic (r=blank); probabilistic (r=p); and veristic (r=v). Preponderantly, the constraints are possibilistic. 47 /152LAZ 8/6/2010

48. EXAMPLES: POSSIBILISTIC Robert is tallHeight(Robert) is tall Most Swedes are tall Count(tall.Swedes/Swedes) is most XR blank X R 48 /152LAZ 8/6/2010

49. 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 ) 49 /152LAZ 8/6/2010

50. 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] 50 /152LAZ 8/6/2010

51. IMPORTANT POINTS In representation of p as a generalized constraint, p: X isr R, there are two important points that have to be noted. First, X need not be a scalar variable. X may be vector-valued or, more generally, have the structure of a semantic network. 51 /152LAZ 8/6/2010

52. CONTINUED For example, in the case of the proposition, p: Robert gave a ring to Anne, X may be represented as the 3- tuple (Giver, Recipient, Object), with the corresponding values of R being (Robert, Anne, Ring). 52 /152LAZ 8/6/2010

53. CONTINUED Second, in general, X is not unique. However, it is usually the case that among possible choices either there is one that has higher plausibility than others, or there are a few that are closely related. For example, if p: Leslie is much taller than Ixel then a plausible choice of X is X: Height(Leslie) 53 /152LAZ 8/6/2010

54. CONTINUED in which case the corresponding constraining relation is R: Much taller than Ixel. Another plausible choice is X: (Height(Leslie); Height(Ixel)). Correspondingly, R: Much taller 54 /152LAZ 8/6/2010

55. CONTINUED With regard to the third question, the constraint in the proposition: Robert is tall, is possibilistic in the sense that it defines possible values of Height (Robert), with the understanding that possibility is a matter of degree. 55 /152LAZ 8/6/2010

56. CANONICAL FORM of p: CF(p) When the meaning of p is represented as a generalized constraint, the expression X isr R is referred to as the canonical form of p, CF(p). Thus, CF(p): X isr R The concept of a canonical form of p has a position of centrality in precisiation of meaning of p. 56 /152LAZ 8/6/2010

57. NOTE It is important to note that the use of generalized constraints in precisiation of propositions drawn from a natural language is greatly facilitated by the fact that, as noted earlier, natural language constraints are for the most part possibilistic—and hence are easy to manipulate. 57 /152LAZ 8/6/2010

58. THE CONCEPT OF EXPLANATORY DATABASE (ED) In generalized-constraint-based semantics, the concept of an explanatory database, ED, serves as a basis for precisiation of meaning of p. (Zadeh 1984) More concretely, ED is a collection of relations, with the names of relations drawn, but not exclusively, from the constituents of p.1984 58 /152LAZ 8/6/2010

59. CONTINUED Basically, ED may be viewed as the information which is needed to define X and R. For example, for the proposition, p: Most Swedes are tall, ED may be represented as: ED=POPULATION.SWEDES[Name; Height]+TALL[Height;µ]+ MOST[Proportion;µ], where + plays the role of comma. 59 /152LAZ 8/6/2010

60. CONTINUED It is important to note that definition of X and R may be viewed as precisiation of X and R. Precisiation of X and R is needed because X and R are described in a natural language. In relation to possible world semantics, ED may be viewed as the description of a possible world. 60 /152LAZ 8/6/2010

61. ADDITIONAL EXAMPLE OF ED p: Brian is much taller than most of his friends. X: Height of Brian. R: Much taller than most of his friends. ED=HEIGHT[Name; Height]+ FRIENDS.BRIAN[Name; µ]+ MUCH.TALLER [Height1; Height2; µ]+ MOST[Proportion; µ] 61 /152LAZ 8/6/2010

62. CONTINUED In FRIENDS.BRIAN, µ is the degree to which Name is a friend of Brian. 62 /152LAZ 8/6/2010

63. NOTE It is important to note that relations in ED are uninstantiated, that is, the values of database variables, v 1, …, v n —entries in relations in ED—are not specified. 63 /152LAZ 8/6/2010

64. THE CONCEPT OF A PRECISIATED CANONICAL FORM, CF*(p) After X and R have been identified and the explanatory database, ED, has been constructed, X and R may be defined as functions of ED, that is, functions of database variables. As was noted earlier, definition of X and R may be viewed as precisiation of X and R. Precisiated X and R are denoted as X* and R*, respectively. 64 /152LAZ 8/6/2010

65. CONTINUED A canonical form, CF*(p), with precisiated values of X and R, X* and R*, will be referred to as a precisiated canonical form. In the following, construction of the precisiated canonical form of p is discussed in greater detail. 65 /152LAZ 8/6/2010

66. 66 /152LAZ 8/6/2010

67. The concepts discussed so far provide a basis for a relatively straightforward procedure for constructing the precisiated canonical form of a given proposition, p. The precisiated canonical form may be viewed as a computational model of p. Effectively, the precisiated canonical form may be interpreted as a representation of precisiated meaning of p. 67 /152LAZ 8/6/2010

68. A summary of the procedure for computing the precisiated canonical form of p is presented in the following. 68 /152LAZ 8/6/2010

69. PROCEDURE Step 1. Clarification The first step is clarification, if needed, of the meaning of p. This step requires world knowledge. Examples: Overeating causes obesity Most of those who overeat are obese. Obesity is caused by overeating Most of those who are obese, overeat. clarification 69 /152LAZ 8/6/2010

70. CONTINUED Young men like young women Most young men like mostly young women. Swedes are much taller than Italians Most Swedes are much taller than most Italians. Step 2. Identification (explicitation) of X and R. Identify the constrained variable, X, and the corresponding constraining relation, R. clarification 70 /152LAZ 8/6/2010

71. CONTINUED Step 3. Construction of ED. What information is needed—but not necessarily minimally—to precisiate (define) X and R? An answer to this question identifies the explanatory database, ED. Equivalently, ED may be viewed as an answer to the question: What information is needed—but not necessarily minimally—to compute the truth-value of p? 71 /152LAZ 8/6/2010

72. CONTINUED Step 4. Precisiation of X and R. How can the information in ED be used to precisiate the values of X and R? This step leads to precisiated values of X and R, X* and R*, and thus results in the precisiated canonical form, CF*(p). Precisiated X* and R* may be expressed as functions of ED and, more specifically, as functions of database variables, v 1, …, v n. 72 /152LAZ 8/6/2010

73. A KEY POINT It is important to observe that in the case of possibilistic constraints, CF*(p) induces a possibilistic constraint on database variables, v 1, …, v n, in ED. This constraint may be interpreted as the possibility distribution of database variables in ED or, equivalently, as a possibility distribution on the state space, SS(p), of p—a possibility distribution which is induced by p. The possibility distribution induced by p may be viewed as the intension of p. 73 /152LAZ 8/6/2010

74. CONTINUED Step 5. (Optional) Computation of truth- value of p. The truth-value of p depends on ED. The truth-value of p, t(p, ED), may be computed by assessing the degree to which the generalized constraint, X* isr R*, is satisfied. It is important to observe that the possibility of an instantiated ED given p is equal to the truth value of p given instantiated ED (Zadeh 1981).1981 End of procedure. 74 /152LAZ 8/6/2010

75. NOTE It is important to note that humans have no difficulty in learning how to use the procedure. The principal reason is: Humans have world knowledge. It is hard to build world knowledge into machines. 75 /152LAZ 8/6/2010

76. SUMMARY The generalized constraint on X*, GC(X*), induces (converts into) a generalized constraint, GC(V), on the database variables, V=(v 1, …, v n ). For possibilistic constraints, GC(V) may be expressed as: f(V) is A where f is a function of database variables and A is a fuzzy relation (set) in the space of database variables. p X* is R* GC(V) GC(X*) precisiationconversion 76 /152LAZ 8/6/2010

77. EXAMPLE Note. In the following example r=blank, that is, the generalized constraints are possibilistic. 1. p: Most Swedes are tall Step 1. Clarification. Clarification not needed Step 2. Identification (explicitation) of X and R. X is identified as the proportion of tall Swedes among Swedes. 77 /152LAZ 8/6/2010

78. CONTINUED Correspondingly, R is identified as Most. Digression. In fuzzy logic, proportion is defined as a relative ΣCount. (Zadeh 1983) More specifically, if A and B are fuzzy sets in U, U={u 1, …, u n }, the ΣCount(cardinality) of A is defined as:1983 78 /152LAZ 8/6/2010

79. CONTINUED The relative ΣCount of B in A is defined as: where =intersection and =min 79 /152LAZ 8/6/2010

80. CONTINUED In application to the example under consideration, assume that the height of ith Swede, Name i, is h i and that the grade of membership of h i in tall is µ tall (h i ), i=1, …, n. X may be expressed as: Step 3. Construction of ED. The needed information is contained in the explanatory database, ED, where 80 /152LAZ 8/6/2010

81. CONTINUED ED= POPULATION.SWEDES[Name; Height]+ TALL[Height; µ]+ MOST[Proportion; µ] Step 4. Precisiation of X and R. In relation to ED, precisiated X and R may be expressed as: R* = MOST[Proportion; µ] 81 /152LAZ 8/6/2010

82. CONTINUED The precisiated canonical form is expressed as: CF*p=X* is R* where R* = MOST[Proportion; µ] 82 /152LAZ 8/6/2010

83. CONTINUED Step 5. The truth-value of p, t(p, ED), is the degree to which the constraint in Step 4 is satisfied. More concretely, Note. The right-hand side of this equation may be viewed as a constraint on database variables h 1, …, h n, µ tall and µ most. 83 /152LAZ 8/6/2010

84. SUMMATION Natural languages are pervasively imprecise, especially in the realm of meaning. The primary source of imprecision is unsharpness of class boundaries. In this sense, words, phrases, propositions and commands in natural languages are preponderantly imprecise. Precisiation of meaning is a prerequisite to computation. 84 /152LAZ 8/6/2010

85. 85 /152LAZ 8/6/2010

86. FROM PRECISIATION TO COMPUTATION Phase 1 precisiation p 1. p n-1 P n q p 1 *. p n-1 * p n * q* I Phase 2 (Granular Computing) computation with generalized constraints X 1 isr 1 R 1. X n-1 isr n-1 R n-1 X n isr n R n q* I* Ans(q/I) : X 1 isr 1 R 1 : X n-1 isr n-1 R n-1 : X n isr n R n I* 86 /152LAZ 8/6/2010

87. CONTINUED A generalized constraint may be viewed as a representation of a granule. Computation with generalized constraints involves granular computing. 87 /152LAZ 8/6/2010

88. KEY POINTS Representation of propositions drawn from a natural language as generalized constraints opens the door to computation with information described in a natural language. In large measure, computation with generalized constraints involves the use of rules which govern propagation and counterpropagation of generalized constraints. Among such rules, the principal rule is the Extension Principle (Zadeh 1965, 1975 a, b and c).1965abc 88 /152LAZ 8/6/2010

89. EXTENSION PRINCIPLE (POSSIBILISTIC) X is a variable which takes values in U, and f is a function from U to V. The point of departure is a possibilistic constraint on f(X) expressed as f(X) is A, where A is a fuzzy set in V which is defined by its membership function μ A (v), vεV. g is a function from U to W. The possibilistic constraint on f(X) induces a possibilistic constraint on g(X) which may be expressed as g(X) is ?B, where B is a fuzzy set in W. The question is: What is B? In symbols, 89 /152LAZ 8/6/2010

90. CONTINUED f(X) is A g(X) is ?B subject to where µ A and µ B are the membership functions of A and B, respectively. The answer to this question is the solution of a mathematical program expressed as: 90 /152LAZ 8/6/2010

91. A B f -1 f g propagation U V counterpropagation W g(f -1 (A)) f -1 (A) µ A (f(u)) f(u) u w STRUCTURE OF THE EXTENSION PRINCIPLE 91 /152LAZ 8/6/2010

92. CW—BASIC COMPUTATIONAL PROCESS p XRXR identification ED construction X* R* precisiation GC(X*) f(V) is A conversion q q* precisiation GC(V) g(V) is ?B conversion extension principle f(V) is A g(V) is ?B Ans(q/p) 92 /152LAZ 8/6/2010

93. 93 /152LAZ 8/6/2010

94. INFORMAL EXPOSITION OF GCS— CLARIFICATION DIALOGUE The basic ideas which underlie precisiation of meaning and, more particularly, generalized-constraint- based semantics, are actually quite simple. To bring this out, it is expedient to supplement a formal exposition of GCS with an informal narrative in the form of a dialogue between Robert and Lotfi. In large measure, the narrative is self-contained. 94 /152LAZ 8/6/2010

95. DIALOGUE Robert: Lotfi, generalized-constraint- based semantics looks complicated to me. Can you explain in simple terms the basic ideas which underlie GCS? 95 /152LAZ 8/6/2010

96. CONTINUED Lotfi: I will be pleased to do so. Let us start with an example, p: Most Swedes are tall. p is a proposition. As a proposition, p is a carrier of information. Without loss of generality, we can assume that p is a carrier of information about a variable, X, which is implicit in p. If I asked you what is this variable, what would you say? 96 /152LAZ 8/6/2010

97. CONTINUED Robert: As I see it, p tells me something about the proportion of tall Swedes among Swedes. Lotfi: Right. What does p tell you about the value of the variable? Robert: To me, the value is not sharply defined. I would say it is fuzzy. Lotfi: So what is it? Robert: It is the word “most.” 97 /152LAZ 8/6/2010

98. CONTINUED Lotfi: You are right. So what we see is that p may be interpreted as the assignment of a value “most” to the variable, X: Proportion of tall Swedes among Swedes. 98 /152LAZ 8/6/2010

99. CONTINUED As you can see, a basic difference between a proposition drawn from a natural language and a proposition drawn from a mathematical language is that in the latter the variables and the values assigned to them are explicit, whereas in the former the variables and the assigned values are implicit. 99 /152LAZ 8/6/2010

100. CONTINUED There is an additional difference. When p is drawn from a natural language, the assigned value is not sharply defined— typically it is fuzzy, as “most” is. When p is drawn from a mathematical language, the assigned value is sharply defined. Robert: I get the idea. So what comes next? 100 /152LAZ 8/6/2010

101. CONTINUED Lotfi: There is another important point. When p is drawn from a natural language, the value assigned to X is not really a value of X—it is a constraint (restriction) on the values which X is allowed to take. This suggests an unconventional definition of a proposition, p, drawn from a natural language. Specifically, a proposition is an implicit constraint on an implicit variable. 101 /152LAZ 8/6/2010

102. CONTINUED I should like to add that the constraints which I have in mind are not standard constraints—they are so-called generalized constraints. 102 /152LAZ 8/6/2010

103. CONTINUED Robert: What is a generalized constraint? Why do we need generalized constraints? Lotfi: A generalized constraint is expressed as: X isr R 103 /152LAZ 8/6/2010

104. CONTINUED where X is the constrained variable, R is the constraining relation—typically a fuzzy set—and r is an indexical variable which defines how R constrains X. Let me explain why the concept of a generalized constraint is needed in precisiation of meaning of a proposition drawn from a natural language. 104 /152LAZ 8/6/2010

105. CONTINUED Standard constraints are hard in the sense that they have no elasticity. In a natural language, meaning can be stretched. What this implies is that to represent meaning, a constraint must have elasticity. To deal with richness of meaning, elasticity is necessary but not sufficient. Consider the proposition: Usually most flights leave on time. 105 /152LAZ 8/6/2010

106. CONTINUED What is the constrained variable and what is the constraining relation in this proposition? Actually, for most propositions drawn from a natural language a large repertoire of constraints is not necessary. What is sufficient are three so-called primary constraints and their combinations. The primary constraints are: possibilistic, probabilistic and veristic. 106 /152LAZ 8/6/2010

107. CONTINUED Here are simple examples of primary constraints: Possibilistic constraint: Robert is possibly French and possibly German Probabilistic constraint: With probability 0.75 Robert is German With probability 0.25 Robert is French Veristic constraint: Robert is three-quarters German and one-quarter French 107 /152LAZ 8/6/2010

108. CONTINUED The role of primary constraints is analogous to the role of primary colors: red, green and blue. In most cases, constraints are possibilistic. Possibilistic constraints are much easier to manipulate than probabilistic constraints. 108 /152LAZ 8/6/2010

109. CONTINUED Robert: Could you clarify what you have in mind when you talk about elasticity of meaning? Lotfi: I admit that I did not say enough. Let me elaborate. In a natural language, meaning can be stretched. Consider a simple example, Robert is young. Assume that young is a fuzzy set and Robert is 30. 109 /152LAZ 8/6/2010

110. CONTINUED Furthermore, assume that in a particular context the grade of membership of 30 in young is 0.8. To apply young to Robert, the meaning of young must be stretched. To what degree? In fuzzy logic, the degree of stretch is equated to (1 - grade of membership of 30 in young.) Thus, the degree of stretch is 0.2. 110 /152LAZ 8/6/2010

111. CONTINUED Furthermore, the grade of membership of 30 in young is interpreted as the possibility that Robert is 30, given that Robert is young. What this implies is that the fuzzy set young defines the possibility distribution of the variable Age (Robert). Note that the fuzzy set young is a restriction on the values which the variable Age (Robert) can take. 111 /152LAZ 8/6/2010

112. CONTINUED It is in this sense that the proposition Robert is young is a possibilistic constraint on Age (Robert). Now, in a natural language almost all words and phrases are labels of fuzzy sets. What this means is that in a natural language the meaning of words and phrases can be stretched, as in the Robert example. 112 /152LAZ 8/6/2010

113. CONTINUED It is in this sense that words and phrases in a natural language have elasticity. Another important point. What I have said so far explains why in the realm of natural languages most constraints are possibilistic. This is equivalent to saying what I said already, namely, that in a natural language most words and phrases are labels of fuzzy sets. 113 /152LAZ 8/6/2010

114. CONTINUED Robert: Many thanks. You clarified what was not clear to me. 114 /152LAZ 8/6/2010

115. CONTINUED Lotfi: May I add that there is an analogy that may be of assistance. More specifically, the fuzzy set young may be represented as a chain linked to a spring, as shown in the next viewgraph. The left end of the chain is fixed and the position of the right end of the spring represents the value of the variable Age (Robert). 115 /152LAZ 8/6/2010

116. CONTINUED The force that is applied to the right end of the spring is a measure of grade of membership. Initially, the length of the chain is 0, as is the length of the spring. force Age 116 /152LAZ 8/6/2010

117. CONTINUED Robert: Many thanks for the explanation. The analogy helps to understand what you mean by elasticity of meaning. Lotfi: I should like to add that elasticity of meaning is a basic characteristic of natural languages. Elasticity of meaning is a neglected issue in the literatures of linguistics, computational linguistics and philosophy of languages. There is a reason. 117 /152LAZ 8/6/2010

118. CONTINUED Traditional theories of natural language are based on bivalent logic. Bivalent logic, by itself or in combination with probability theory, is not the right tool for dealing with elasticity of meaning. What is needed for this purpose is fuzzy logic. In fuzzy logic everything is or is allowed to be a matter of degree. 118 /152LAZ 8/6/2010

119. CONTINUED Robert: Thanks again for the clarification. Going back to where we left of suppose I figured out what is the constrained variable, X, and the constraining relation, R. Is there something else that has to be done? 119 /152LAZ 8/6/2010

120. CONTINUED Lotfi: Yes, there is. You see, X and R are described in a natural language. What this means is that we are not through with precisiation of meaning of p. What remains to be done is precisiation (definition) of X and R. 120 /152LAZ 8/6/2010

121. CONTINUED For this purpose, we construct a so- called explanatory database, ED, which consists of a collection of relations in terms of which X and R can be defined. The entries in relations in ED are referred to as database variables. Unless stated to the contrary, database variables are assumed to be uninstantiated. 121 /152LAZ 8/6/2010

122. CONTINUED Robert: Can you be more specific? Lotfi: To construct ED you ask yourself the question: What information—in the form of a collection or relations—is needed to precisiate (define) X and R? Looking at p, we see that to precisiate X we need two relations: POPULATION.SWEDES[Name; Height] and TALL[Height; µ]. 122 /152LAZ 8/6/2010

123. CONTINUED In the relation TALL[Height; µ], µ is the grade of membership of a value of Height, h, in the fuzzy set tall. So far as R is concerned, the needed relation is MOST[Proportion; µ], where µ is the grade of membership of a value of Proportion in the fuzzy set Most. 123 /152LAZ 8/6/2010

124. CONTINUED Equivalently, it is frequently helpful to ask the question: What is the information which is needed to assess the degree to which p is true? 124 /152LAZ 8/6/2010

125. CONTINUED At this point, we can express ED as the collection: ED= POPULATION.SWEDES[Name; Height]+ TALL[Height; µ]+ MOST[Proportion; µ] in which for convenience plus is used in place of comma. 125 /152LAZ 8/6/2010

126. CONTINUED Robert: So, we have constructed ED for the proposition, p: Most Swedes are tall. More generally, given a proposition, p, how difficult is it to construct ED for p? Lotfi: For humans it is easy. A few examples suffice to learn how to construct ED. Construction of ED is easy for humans because humans have world knowledge. At this juncture, we do not have an algorithm for constructing ED. 126 /152LAZ 8/6/2010

127. CONTINUED Robert: Now that we have ED, what comes next? Lotfi: We can use ED to precisiate (define) X and R. Let us start with X. In words, X is described as the proportion of tall Swedes among Swedes. Let us assume that in the relation POPULATION.SWEDES there are n names. Then the proportion of tall Swedes among Swedes would be the number of tall Swedes divided by n. 127 /152LAZ 8/6/2010

128. CONTINUED Here we come to a problem. Tall Swedes is a fuzzy subset of Swedes. The question is: What is the number of elements in a fuzzy set? In fuzzy logic, there are different ways of answering this question. The simplest is referred to as the ΣCount. More concretely, if A is a fuzzy set with a membership function µ A, then the ΣCount of A is defined as the sum of grades of membership in A. 128 /152LAZ 8/6/2010

129. CONTINUED In application to the number of tall Swedes, the ΣCount of tall Swedes may be expressed as: ΣCount(tall.Swedes)= where h i is the height of Name i. Consequently, the proportion of tall Swedes among Swedes may be written as: 129 /152LAZ 8/6/2010

130. CONTINUED This expression may be viewed as a precisiation (definition) of X in terms of ED. More specifically, X is expressed as a function of database variables h 1, …, h n, µ tall and µ most. Precisiation (definition) of R is simpler. Specifically, R=Most, where Most is a fuzzy set. At this point, we have precisiated (defined) X and R in terms of ED. 130 /152LAZ 8/6/2010

131. CONTINUED Robert: So what have we accomplished? Lotfi: We started with a proposition, p: Most Swedes are tall. We interpreted p as a generalized (possibilistic) constraint. We identified the constrained variable, X, as the proportion of tall Swedes among Swedes. We identified the constraining relation, R, as a fuzzy set, Most. Next, we constructed an explanatory database, ED. 131 /152LAZ 8/6/2010

132. CONTINUED Finally, we precisiated (defined) X, R and q in terms of ED, that is, as function of database variables h 1, …, h n, µ tall and µ most. In this way, we precisiated the meaning of p, which was our objective. The precisiated meaning may be expressed as the constraint: Robert: So, you precisiated the meaning of p. What purpose does it serve? is Most 132 /152LAZ 8/6/2010

133. CONTINUED Lotfi: The principal purpose is the following. Unprecisiated (raw) propositions drawn from a natural language cannot be computed with. Precisiation is a prerequisite to computation. What is important to understand is that precisiation of meaning opens the door to computation with natural language. 133 /152LAZ 8/6/2010

134. CONTINUED Robert: Sounds great. I am impressed. However, it is not completely clear to me what you have in mind when you say “opens the door to computation with natural language.” Can you clarify it? Lotfi: With pleasure. Computation with natural language or, more or less equivalently, Computing with Words (CW or CWW), is largely unrelated to natural language processing. 134 /152LAZ 8/6/2010

135. CONTINUED More specifically, computation with natural language is focused on computation with information described in a natural language. Typically, what is involved is solution of a problem which is stated in a natural language. Let me go back to our example, p: Most Swedes are tall. Given this information, how can you compute the average height of Swedes? 135 /152LAZ 8/6/2010

136. CONTINUED Robert: Frankly, your question makes no sense to me. Are you serious? How can you expect me to compute the average height of Swedes from the information that most Swedes are tall? Lotfi: That is conventional wisdom. A mathematician would say that the problem is ill-posed. It appears to be ill- posed for two reasons. 136 /152LAZ 8/6/2010

137. CONTINUED First, because the given information: Most Swedes are tall, is fuzzy, and second, because you assume that I am expecting you to come up with a crisp answer like “the average height of Swedes is 5’ 10.” Actually, what I expect is a fuzzy answer—it would be unreasonable to expect a crisp answer. Robert: Thanks for the clarification. I am beginning to see the point of your question. 137 /152LAZ 8/6/2010

138. CONTINUED Lotfi: I should like to add a key point. The problem becomes well-posed if p is precisiated. This is the essence of Computing with Words. 138 /152LAZ 8/6/2010

139. CONTINUED Robert: I am beginning to understand the need for precisiation, but my understanding is not complete as yet. Can you explain how the average height of Swedes can be computed from precisiated p? Lotfi: Recall that precisiated p is a possibilistic constraint expressed as: is Most 139 /152LAZ 8/6/2010

140. CONTINUED From the definition of a possibilistic constraint it follows that the constraint on X may be rewritten as: What this expression means is that given the h i, µ tall and µ most, we can compute the degree, t, to which the constraint is satisfied. 140 /152LAZ 8/6/2010

141. CONTINUED It is this degree, t, that is the truth-value of p. Now, here is a key idea. The precisiated p constrains X. X is a function of database variables. It follows that indirectly p constrains database variables. This has important implications. Let me elaborate. 141 /152LAZ 8/6/2010

142. CONTINUED What we see is that the constraint induced by p on the h i is of the general form f(h 1, …, h n ) is Most What we are interested in is the induced constraint on the average height of Swedes. The average height of Swedes may be expressed as: 142 /152LAZ 8/6/2010

143. CONTINUED This expression is of the general form g(h 1, …, h n ) is ?h ave where ?h ave is a fuzzy set that we want to compute. 143 /152LAZ 8/6/2010

144. CONTINUED At this stage, we can employ the Extension Principle of fuzzy logic to compute h ave. (Zadeh 1975 I, II & III) In general terms, this principle tells us that from a given possibilistic constraint of the formIIIIII f(x 1, …, x n ) is A in which A is a fuzzy set, we can derive an induced possibilistic constraint on g(x 1, …, x n ), g(x 1, …, x n ) is ?B, 144 /152LAZ 8/6/2010

145. CONTINUED in which B is a fuzzy set defined by the solution of the mathematical program µ B (v)=sup x 1, …, x n µ A (f(x 1, …, x n )) subject to v=g(x 1, …, x n ) In application to our example, what we see is that we have reduced computation of the average height of Swedes to the solution of the mathematical program 145 /152LAZ 8/6/2010

146. CONTINUED µ B (v)=sup h 1, …, h n µ most (f(h 1, …, h n )) subject to In effect, this is the solution to the problem which I posed to you. As you can see, reduction of the original problem to the solution of a mathematical program is not so simple. 146 /152LAZ 8/6/2010

147. CONTINUED However, solution of the mathematical program to which the original problem is reduced, is well within the capabilities of desktop computers. 147 /152LAZ 8/6/2010

148. CONTINUED Robert: I am beginning to see the basic idea. Through precisiation, you have reduced the problem of computation with information described in a natural language—a seemingly ill-posed problem—to a well-posed tractable problem in mathematical programming. I am impressed by what you have accomplished, though I must say that the reduction is nontrivial. 148 /152LAZ 8/6/2010

149. CONTINUED Without your explanation, it would be hard to see the basic ideas. I can also see why computation with natural language is a move into a new and largely unexplored territory. Thank you for clarifying the import of your statement: precisiation of meaning opens the door to computation with natural language. 149 /152LAZ 8/6/2010

150. CONTINUED Lotfi: I appreciate your comment. May I add that I believe—but have not verified it as yet—that in closed form the solution to the mathematical program may be expressed as: h ave is  Most × Tall where Most × Tall is the product of fuzzy numbers Most and Tall. Robert: This is a very interesting result, if true. It agrees with my intuition. 150 /152LAZ 8/6/2010

151. CONTINUED Lotfi: I appreciate your comment. I would like to conclude our dialogue with a prediction. As we move further into the age of machine intelligence and automated reasoning, the complex of problems related to computation with information described in a natural language, is certain to grow in visibility and importance. 151 /152LAZ 8/6/2010

152. CONTINUED The informal dialogue between Robert and Lotfi has come to an end. 152 /152LAZ 8/6/2010