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Granular Computing—Computing with Uncertain, Imprecise and Partially True Data Lotfi A. Zadeh

Granular Computing—Computing with Uncertain, Imprecise and Partially True Data Lotfi A. Zadeh Computer Science Division Department of EECS UC Berkeley ISSDQ’07 Enschede, The Netherlands June13, 2007 URL: http://www-bisc.cs.berkeley.edu URL: http://www.cs.berkeley.edu/~zadeh/

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Granular Computing—Computing with Uncertain, Imprecise and Partially True Data Lotfi A. Zadeh

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  1. Granular Computing—Computing with Uncertain, Imprecise and Partially True Data Lotfi A. Zadeh Computer Science Division Department of EECSUC Berkeley ISSDQ’07 Enschede, The Netherlands June13, 2007 URL: http://www-bisc.cs.berkeley.edu URL: http://www.cs.berkeley.edu/~zadeh/ Email: Zadeh@eecs.berkeley.edu LAZ 6/5/2007

  2. PREAMBLE LAZ 6/5/2007

  3. GRANULAR COMPUTING (GrC) Information is the life blood of modern society. Decisions are based on information. More often than not, decision-relevant information is imperfect in the sense that it is in part imprecise and/or uncertain and/or incomplete and/or conflicting and/or partially true. There is a long list of methods for dealing with imperfect information. Included in this list are probability theory, possibility theory, fuzzy logic, Dempster-Shafer theory, rough set theory and granular computing. Rough set theory and granular computing are relatively recent listings. LAZ 6/5/2007

  4. CONTINUED Existing methods, based as they are on bivalent logic and bivalent-logic-based probability theory, have serious limitations. Granular computing, which is based on fuzzy logic, substantially enhances our ability to reason, compute and make decisions based on imperfect information. Use of granular computing is a necessity in dealing with imprecise probabilities. LAZ 6/5/2007

  5. WHAT IS FUZZY LOGIC? There are many misconceptions about fuzzy logic. To begin with, fuzzy logic is not fuzzy. In large measure, fuzzy logic is precise. Another source of confusion is the duality of meaning of fuzzy logic. In a narrow sense, fuzzy logic is a logical system. But in much broader sense that is in dominant use today, fuzzy logic, or FL for short, is much more than a logical system. More specifically, fuzzy logic has many facets. There are four principal facets. LAZ 6/5/2007

  6. FACETS OF FUZZY LOGIC (a) the logical facet, FLl; (b) the fuzzy-set-theoretic facet, FLs; (c) the epistemic facet, FLe; and (d) the relational facet, FLr. logical (narrow sense) FLIs G/G FLr fuzzy-set-theoretic relational FL epistemic FLe G/G: Graduation/Granulation LAZ 6/5/2007

  7. GRADUATION AND GRANULATION The basic concepts of graduation and granulation form the core of FL and are the principal distinguishing features of fuzzy logic. More specifically, in fuzzy logic everything is or is allowed to be graduated, that is, be a matter of degree or, equivalently, fuzzy. Furthermore, in fuzzy logic everything is or is allowed to be granulated, with a granule being a clump of attribute-values drawn together by indistinguishability, similarity, proximity or functionality. LAZ 6/5/2007

  8. GRADUATION AND GRANULATION middle-aged µ µ old young 1 1 0 Age 0 quantized Age granulated For example, Age is granulated when its values are described as young, middle-aged and old. A linguistic variable may be viewed as a granulated variable whose granular values carry linguistic labels. In an informal way, graduation and granulation play pivotal roles in human cognition. LAZ 6/5/2007

  9. MODALITIES OF VALUATION granular valuation: assignment of a value to a variable numerical: Vera is 48 linguistic: Vera is middle-aged Computing with Words (CW): Vera is likely to be middle-aged NL-Computation: Vera has a teenager son and a daughter in mid-twenties world knowledge: child-bearing age ranges from about 16 to about 42. LAZ 6/5/2007

  10. GRANULATION—A CORE CONCEPT RST rough set theory NL-C CTP granulation NL-Computation computational theory of perceptions granular computing GrC Granular Computing= ballpark computing LAZ 6/5/2007

  11. GRANULATION RST GRC f-granulation c-granulation granulation: partitioning (crisp or fuzzy) of an object into a collection of granules, with a granule being a clump of elements drawn together by indistinguishability, equivalence, similarity, proximity or functionality. example: Body head+neck+chest+arms+···+feet. Set partition into equivalence classes LAZ 6/5/2007

  12. granule L M S 0 S M L GRANULATION OF A FUNCTION GRANULATION=SUMMARIZATION Y f 0 Y medium × large *f (fuzzy graph) perception f *f : 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 X 0 LAZ 6/5/2007

  13. GRANULATION OF A PROBABILITY DISTRIBUTION X is a real-valued random variable probability P3 g P2 P1 X 0 A2 A1 A3 BMD: 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 6/5/2007

  14. GRANULAR VS. GRANULE-VALUED DISTRIBUTIONS g(u): probability density of X P distribution P1 P2 P Pn p1 pn 0 X … A1 A2 A An granules LAZ 6/5/2007

  15. PRINCIPAL TYPES OF GRANULES • Possibilistic • X is a number in the interval [a, b] • Probabilistic • X is a normally distributed random variable with mean a and variance b • Veristic • X is all numbers in the interval [a, b] • Hybrid • X is a random set LAZ 6/5/2007

  16. SINGULAR AND GRANULAR VALUES • X is a variable taking values in U • a, aεU, is a singular value of X if a is a singleton • A is a granular value of X if A is a granule, that is, A is a clump of values of X drawn together by indistinguishability, equivalence, similarity, proximity or functionality. • A may be interpreted as a representation of information about a singular value of X. • A granular variable is a variable which takes granular values • A linguistic variable is a granular variable with linguistic labels of granular values. LAZ 6/5/2007

  17. SINGULAR AND GRANULAR VALUES A granular value of X a singular value of X universe of discourse singular granular unemployment temperature blood pressure LAZ 6/5/2007

  18. ATTRIBUTES OF A GRANULE • Probability measure • Possibility measure • Verity measure • Length • Volume • … LAZ 6/5/2007

  19. RATIONALES FOR GRANULATION granulation imperative (forced) intentional (deliberate) • value of X is not known precisely value of X need not be known precisely Rationale 1 Rationale 2 Rationale 2: precision is costly if there is a tolerance for imprecision, exploited through granulation of X LAZ 6/5/2007

  20. CLARIFICATION—THE MEANING OF PRECISION PRECISE v-precise m-precise • precise value • p: X is a Gaussian random variable with mean m and variance 2. m and 2 are precisely defined real numbers • p is v-imprecise and m-precise • p: X is in the interval [a, b]. a and b are precisely defined real numbers • p is v-imprecise and m-precise precise meaning granulation = v-imprecisiation LAZ 6/5/2007

  21. MODALITIES OF m-PRECISIATION m-precisiation mh-precisiation mm-precisiation machine-oriented human-oriented mm-precise: mathematically well-defined LAZ 6/5/2007

  22. CLARIFICATION young 1 0 Rationale 2: if there is a tolerance for imprecision, exploited through granulation of X Rationale 2: if there is a tolerance for v-imprecision, exploited through granulation of X followed by mm-precisiation of granular values of X Example: Lily is 25 Lily is young LAZ 6/5/2007

  23. RATIONALES FOR FUZZY LOGIC RATIONALE 1 IDL v-imprecise mm-precisiation • BL: bivalent logic language • FL: fuzzy logic language • NL: natural language • IDL: information description language • FL is a superlanguage of BL • Rationale 1: information about X is described in FL via NL LAZ 6/5/2007

  24. RATIONALES FOR FUZZY LOGIC RATIONALE 2—Fuzzy Logic Gambit v-precise v-imprecise v-imprecisiation mm-precisiation Fuzzy Logic Gambit: if there is a tolerance for imprecisiation, exploited by v-imprecisiation followed by mm-precisiation • Rationale 2 plays a key role in fuzzy control LAZ 6/5/2007

  25. CHARACTERIZATION OF A GRANULE IDL bivalent logic fuzzy logic natural language information = generalized constraint granular value of X = information, I(X), about the singular value of X I(X) is represented through the use of an information description language, IDL. BL: SCL (standard constraint language) FL: GCL (generalized constraint language) NL: PNL (precisiated natural language) LAZ 6/5/2007

  26. EXAMPLE—PROBABILISTIC GRANULE Implicit characterization of a probabilistic granule via natural language X is a real-valued random variable Probability distribution of X is not known precisely. What is known about the probability distribution of X is: (a) usually X is much larger than approximately a; usually X is much smaller than approximately b. In this case, information about X is mm-precise and implicit. LAZ 6/5/2007

  27. THE CONCEPT OF A GENERALIZED CONSTRAINT LAZ 6/5/2007

  28. PREAMBLE • In scientific theories, representation of constraints is generally oversimplified. Oversimplification of constraints is a necessity because existing constrained definition languages have a very limited expressive power. The concept of a generalized constraint is intended to provide a basis for construction of a maximally expressive constraint definition language which can also serve as a meaning representation/precisiation language for natural languages. LAZ 6/5/2007

  29. GENERALIZED CONSTRAINT (Zadeh 1986) • Bivalent constraint (hard, inelastic, categorical:) X  C constraining bivalent relation • Generalized constraint on X: GC(X) GC(X): X isr R constraining non-bivalent (fuzzy) relation index of modality (defines semantics) constrained variable r:  | = |  |  |  | … | blank | p | v | u | rs | fg | ps |… bivalent primary • open GC(X): X is free (GC(X) is a predicate) • closed GC(X): X is instantiated (GC(X) is a proposition) LAZ 6/5/2007

  30. CONTINUED • constrained variable • X is an n-ary variable, X= (X1, …, Xn) • 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: (Name1, …, Namen), with each member of the group, Namei, i =1, …, n, associated with an attribute-value, hi, of attribute H. hi may be vector-valued. Symbolically LAZ 6/5/2007

  31. CONTINUED G = (Name1, …, Namen) G[H] = (Name1/h1, …, Namen/hn) G[H is A] = (µA(hi)/Name1, …, µA(hn)/Namen) Basically, G[H] is a relation and G[H is A] is a fuzzy restriction of G[H] Example: tall Swedes Swedes[Height is tall] LAZ 6/5/2007

  32. 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: blank possibilistic constraint; X is R; R is the possibility distribution of X r: v veristic constraint; X isv R; R is the verity distribution of X r: p probabilistic constraint; X isp R; R is the probability distribution of X Standard constraints: bivalent possibilistic, bivalent veristic and probabilistic LAZ 6/5/2007

  33. CONTINUED r: bm bimodal constraint; X is a random variable; R is a bimodal distribution r: rs random set constraint; X isrs R; R is the set- valued probability distribution of X r: fg fuzzy graph constraint; X isfg R; X is a function and R is its fuzzy graph r: u usuality constraint; X isu R means usually (X is R) r: g group constraint; X isg R means that R constrains the attribute-values of the group LAZ 6/5/2007

  34. PRIMARY GENERALIZED CONSTRAINTS • Possibilistic: X is R • Probabilistic: X isp R • Veristic: X isv R • Primary constraints are formalizations of three basic perceptions: (a) perception of possibility; (b) perception of likelihood; and (c) perception of truth • In this perspective, probability may be viewed as an attribute of perception of likelihood LAZ 6/5/2007

  35. STANDARD CONSTRAINTS • Bivalent possibilistic: X  C (crisp set) • Bivalent veristic: Ver(p) is true or false • Probabilistic: X isp R • Standard constraints are instances of generalized constraints which underlie methods based on bivalent logic and probability theory LAZ 6/5/2007

  36. 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 X R X R R X LAZ 6/5/2007

  37. 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] LAZ 6/5/2007

  38. GENERALIZED CONSTRAINT LANGUAGE (GCL) • GCL is an abstract language • GCL is generated by combination, qualification, propagation and counterpropagation of generalized constraints • examples of elements of GCL • X/Age(Monika) is R/young (annotated element) • (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 and counterpropagation LAZ 6/5/2007

  39. EXTENSION PRINCIPLE • The principal rule of deduction in NL-Computation is the Extension Principle (Zadeh 1965, 1975). f(X) is A g(X) is B subject to LAZ 6/5/2007

  40. EXAMPLE • p: most Swedes are tall p*: Count(tall.Swedes/Swedes) is most further precisiation X(h): height density function (not known) X(h)du: fraction of Swedes whose height is in [h, h+du], a  h  b LAZ 6/5/2007

  41. PRECISIATION AND CALIBRATION • µtall(h): membership function of tall (known) • µmost(u): membership function of most (known) height most 1 1 0 0 1 height 0.5 1 fraction X(h) height density function 0 b h (height) a LAZ 6/5/2007

  42. CONTINUED • fraction of tall Swedes: • constraint on X(h) is most granular value LAZ 6/5/2007

  43. DEDUCTION q: What is the average height of Swedes? q*: is ? Q deduction: is most is ? Q LAZ 6/5/2007

  44. THE CONCEPT OF PROTOFORM • Protoform= abbreviation of prototypical form summarization abstraction generalization Pro(p) p p: object (proposition(s), predicate(s), question(s), command, scenario, decision problem, ...) Pro(p): protoform of p Basically, Pro(p) is a representation of the deep structure of p LAZ 6/5/2007

  45. EXAMPLE • p: most Swedes are tall abstraction p Q A’s are B’s generalization Q A’s are B’s Count(G[H is R]/G) is Q LAZ 6/5/2007

  46. EXAMPLES Monika is much younger than Robert (Age(Monika), Age(Robert) is much.younger D(A(B), A(C)) is E gain 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? 2 1 0 option 2 option 1 LAZ 6/5/2007

  47. PROTOFORM EQUIVALENCE object space protoform space • 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 tall Count (A/B) is Q q: Few professors are rich Count (A/B) is Q PF-equivalence class LAZ 6/5/2007

  48. PROTOFORM EQUIVALENCE—DECISION PROBLEM • Pro(backpain)= Pro(surge in Iraq) = Pro(divorce) = Pro(new job)= Pro(new location) • Status quo may be optimal LAZ 6/5/2007

  49. DEDUCTION • In NL-computation, deduction rules are protoformal Example: 1/nCount(G[H is R]) is Q 1/nCount(G[H is S]) is T i µR(hi) is Q i µS(hi) is T µT(v) = suph1, …, hn(µQ(i µR(hi)) subject to v = i µS(hi) values of H: h1, …, hn LAZ 6/5/2007

  50. PROBABILISTIC DEDUCTION RULE Prob {X is Ai} is Pi , i = 1, …, n Prob {X is A} is Q subject to LAZ 6/5/2007

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