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To help protect y our priv acy , PowerPoint prev ented this external picture from being automatically downloaded. To download and display this picture, click Options in the Message Bar, and then click Enable external content.Fuzzy Logic and Fuzzy Systems –Properties & RelationshipsKhurshid Ahmad,Professor of Computer Science,Department of Computer ScienceTrinity College,Dublin-2, IRELANDOctober 3rd, /Teaching.html111
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To help protect y our priv acy , PowerPoint prev ented this external picture from being automatically downloaded. To download and display this picture, click Options in the Message Bar, and then click Enable external content.FUZZY LOGIC & FUZZY SYSTEMSTerminologyFuzzy sets are sets whose elements have degrees ofmembership of the sets.Fuzzy sets are an extension of the classical set.Membership of a set governed by classical settheory is described according to a bivalentcondition — all members of the set definitelybelong to the set whilst all non-members do notbelong to the classical set.Sets governed by the rules of classical set theoryare referred to as crispsets.33
To help protect y our priv acy , PowerPoint prev ented this external picture from being automatically downloaded. To download and display this picture, click Options in the Message Bar, and then click Enable external content.FUZZY LOGIC & FUZZY SYSTEMSBACKGROUND & DEFINITIONSThe concept of a set and set theory are powerful concepts in mathematics. However,the principal notion underlying set theory, that an element can (exclusively) eitherbelong to set or not belong to a set, makes it well nigh impossible to represent much ofhuman discourse. How is one to represent notions like:large profithigh pressuretall manwealthy womanmoderate temperature.Ordinary set-theoretic representations will require the maintenance of a crispdifferentiation in a very artificial manner:high, high to some extent, not quite high, very high etc.44
To help protect y our priv acy , PowerPoint prev ented this external picture from being automatically downloaded. To download and display this picture, click Options in the Message Bar, and then click Enable external content.FUZZY LOGIC & FUZZY SYSTEMSBACKGROUND & DEFINITIONS‘Many decision-making and problem-solving tasks are too complex to beunderstood quantitatively, however, people succeed by using knowledgethat is imprecise rather than precise. Fuzzy set theory, originallyintroduced by Lotfi Zadeh in the 1960's, resembles human reasoning in itsuse of approximate information and uncertainty to generate decisions. Itwas specifically designed to mathematically represent uncertainty andvagueness and provide formalized tools for dealing with the imprecisionintrinsic to many problems. By contrast, traditional computing demandsprecision down to each bit. Since knowledge can be expressed in a morenatural by using fuzzy sets, many engineering and decision problems canbe greatly simplified.’ t.html55
To help protect y our priv acy , PowerPoint prev ented this external picture from being automatically downloaded. To download and display this picture, click Options in the Message Bar, and then click Enable external content.FUZZY LOGIC & FUZZY SYSTEMSBACKGROUND & DEFINITIONSLotfi Zadeh introduced the theory of fuzzy sets: A fuzzy setis a collection of objects that might belong to the set to adegree, varying from 1 for full belongingness to 0 for fullnon-belongingness, through all intermediate valuesZadeh employed the concept of a membership functionassigning to each element a number from the unit intervalto indicate the intensity of belongingness. Zadeh furtherdefined basic operations on fuzzy sets as essentiallyextensions of their conventional ('ordinary') counterparts.Lotdfi Zadeh, Professor in the Graduate School, Computer Science DivisionDepartment of Elec. Eng. and Comp Sciences, University of California Berkeley, CA 94720 -1776Director, Berkeley Initiative in Soft Computing mepages/zadeh.htmlIn 1995, Dr. Zadeh was awarded the IEEE Medal of Honor "For pioneering development of fuzzy logic andits many diverse applications." In 2001, he received the American Computer Machinery’s 2000 AllenNewell Award for seminal contributions to AI through his development of fuzzy logic.66
To help protect y our priv acy , PowerPoint prev ented this external picture from being automatically downloaded. To download and display this picture, click Options in the Message Bar, and then click Enable external content.FUZZY LOGIC & FUZZY SYSTEMSBACKGROUND & DEFINITIONS‘A fuzzy set is a class of objects with a continuum ofgrades of membership. Such a set is characterizedby a membership (characteristic) function whichassigns to each object a grade of membershipranging between zero and one.’ (Zadeh 1965:338)‘The notions of inclusion, union, intersection,complement, relation, convexity, [.] can be extendedto such sets, and various properties of these notionsin the context of fuzzy sets [.] [have been]established.’ (ibid).Lotfi Zadeh (1965). ‘Fuzzy Sets’. Information and Control. Volume 8, Issue 3, June 1965, Pages 338-35377
To help protect y our priv acy , PowerPoint prev ented this external picture from being automatically downloaded. To download and display this picture, click Options in the Message Bar, and then click Enable external content.FUZZY LOGIC & FUZZY SYSTEMSBACKGROUND & DEFINITIONS‘The notion of an event and its probability constitute the most basicconcepts of probability theory. [ ] An event [typically] is a preciselyspecified collection of points in the sample space.’ (Zadeh 1968:421).Consider some everyday events and occurrences:It is a cold day;My computer is approximately 5KG in weight;In 20 tosses of a coin there are several more headsthan tailsIn everyday contexts an event ‘is a fuzzy rather than a sharply definedcollection of points’. Using the concept of a fuzzy set, ‘the notions of anevent and its probability can be extended in a natural fashion to fuzzyevents of the type [described above]’ (ibid)Lotfi Zadeh (1968). ‘Probability Measures of Fuzzy Sets’. J. Mathematical Analysis & Applications Volume23, Issue 2, August 1968, Pages 421-42788
To help protect y our priv acy , PowerPoint prev ented this external picture from being automatically downloaded. To download and display this picture, click Options in the Message Bar, and then click Enable external content.FUZZY LOGIC & FUZZY SYSTEMSBACKGROUND & DEFINITIONS‘In sharp contrast to the idealized world ofmathematics, our perception of the real world ispervaded by concepts which do not have sharplydefined boundaries, e.g., tall, fat, many, most,slowly, old. familiar, relevant, much larger than,kind, etc. A key assumption in fuzzy logic is thatthe denotations of such concepts are fuzzy sets,that is, classes of objects in which the transitionfrom membership to non-membership is gradualrather than abrupt.’ (Zadeh 1990:99).Lotfi Zadeh (1990). ‘Probability Measures of Fuzzy Sets’. International Journal of General Systems. Vol. 17,pp. 95-10599
To help protect y our priv acy , PowerPoint prev ented this external picture from being automatically downloaded. To download and display this picture, click Options in the Message Bar, and then click Enable external content.FUZZY LOGIC & FUZZY SYSTEMSBACKGROUND & DEFINITIONSSystemVariableConventional merical)and tional andRelational Statementsbetween domainobjects A, B:IF A THEN B;A is-a-part-of BA weighs 5KGOrdered sequencesof instructionscomprisingA 5;IF A 5 THENB A 5 Conditional andRelational Statementsbetween domainobjects A, B:IF A (ΨA) THEN B (ΨB)A weighs about 5KGOrdered sequencesof instructionscomprisingA IS-SMALL;IF A IS SMALLTHEN BIS LARGELotfi Zadeh (1990). ‘Probability Measures of Fuzzy Sets’. International Journal of General Systems. Vol. 17,pp. 95-1051010
To help protect y our priv acy , PowerPoint prev ented this external picture from being automatically downloaded. To download and display this picture, click Options in the Message Bar, and then click Enable external content.FUZZY LOGIC & FUZZY SYSTEMSBACKGROUND & DEFINITIONSA FUZZY SYSTEM can be contrasted with aCONVENTIONAL (CRISP) System in three mainways:1.A linguistic variable is defined as a variable whose values are sentences ina natural or artificial language. Thus, if tall, not tall, very tall, very very tall,etc. are values of HEIGHT, then HEIGHT is a linguistic variable.2.Fuzzy conditional statements are expressions of the form IF A THEN B,where A and B have fuzzy meaning, e.g., IF x is small THEN y is large,where small and large are viewed as labels of fuzzy sets.3.A fuzzy algorithm is an ordered sequence of instructions which maycontain fuzzy assignment and conditional statements, e.g., x very small,IF x is small THEN y is large. The execution of such instructions isgoverned by the compositional rule of inference and the rule of thepreponderant alternative.Lotfi Zadeh (1990). ‘Probability Measures of Fuzzy Sets’. International Journal of General Systems. Vol. 17,pp. 95-1051111
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To help protect y our priv acy , PowerPoint prev ented this external picture from being automatically downloaded. To download and display this picture, click Options in the Message Bar, and then click Enable external content.FUZZY LOGIC & FUZZY SYSTEMSPropertiesElements of a fuzzy set maybelong to the set, may not belongto the set, or may belong to adegree.Membership of a crisp set isdescribed by a bivalentcondition; the membership of afuzzy set is described by a multivalent condition.1717
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To help protect y our priv acy , PowerPoint prev ented this external picture from being automatically downloaded. To download and display this picture, click Options in the Message Bar, and then click Enable external content.FUZZY LOGIC & FUZZY SYSTEMSFUZZY SETSAn Example: Consider a set of numbers: X {1, 2, . 10}. Johnny’sunderstanding of numbers is limited to 10, when asked he suggestedthe following. Sitting next to Johnny was a fuzzy logician noting :‘LargeNumber’Comment‘Degree ite poss.’0.87‘Maybe’0.56‘In some cases, not usually’0.25, 4, 3, 2, 1‘Definitely Not’0We can denote Johnny’slarge number’Johnny s notion of ‘largenumber by thefuzzy setA 0/1 0/2 0/3 0/4 0/5 0.2/6 0.5/7 0.8/8 1/919 1/1019
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CONSTRAINT ON THE VALUES THAT MAY BE ASSIGNED TO A VARIABLE. Calculus of Fuzzy Restrictions is essentially a body of concepts and techniques for dealing with fuzzy restrictions in a systematic way: to furnish a conceptual basis for approximate reasoning - neither exact nor inexact reasoning.(cf. Calculus of Probabilities and Probability Theory)
