Lecture 2 Elements of Fuzzy Logic 1 BACKGROUND & DEFINITIONS The concept of a set and set theory are powerful concepts in mathematics. However, the principal notion underlying set theory, that an element can (exclusively) either belong to set or not belong to a set, makes it well nigh impossible to represent much of human discourse. How is one to represent notions like: large profit high pressure tall man wealthy woman moderate temperature. Ordinary set-theoretic representations will require the maintenance of a crisp differentiation in a very artificial manner: high, high to some extent, not quite high, very high etc. BACKGROUND & DEFINITIONS ‘Many decision-making and problem-solving tasks are too complex to be understood quantitatively, however, people succeed by using knowledge that is imprecise rather than precise. Fuzzy set theory, originally introduced by Lotfi Zadeh in the 1960's, resembles human reasoning in its use of approximate information and uncertainty to generate decisions. It was specifically designed to mathematically represent uncertainty and vagueness and provide formalized tools for dealing with the imprecision intrinsic to many problems. By contrast, traditional computing demands precision down to each bit. Since knowledge can be expressed in a more natural by using fuzzy sets, many engineering and decision problems can be greatly simplified.’ http://www.emsl.pnl.gov:2080/proj/neuron/fuzzy/what.html History In the early 1900’s, Lukasiewicz described a three-valued logic. The third value can be translated as the term “possible,” and he assigned it a numeric value between True and False. Later, he explored four-valued logics, five-valued logics, and declared that in principle there was nothing to prevent the derivation of an infinitevalued logic. History Knuth proposed a three-valued logic similar to Lukasiewicz’s. He speculated that mathematics would become even more elegant than in traditional bi-valued logic. His insight was to use the integral range [-1, 0 +1] rather than [0, 1, 2]. History Lotfi Zadeh, at the University of California at Berkeley, first presented multi-valued logic-fuzzy logic in the mid-1960's. Zadeh developed fuzzy logic as a way of processing data. Instead of requiring a data element to be either a member or non-member of a set, he introduced the idea of partial set membership. In 1974 Mamdani and Assilian used fuzzy logic to regulate a steam engine. In 1985 researchers at Bell laboratories developed the first fuzzy logic chip. BACKGROUND & DEFINITIONS Lotfi Zadeh introduced the theory of fuzzy sets: A fuzzy set is a collection of objects that might belong to the set to a degree, varying from 1 for full belongingness to 0 for full non-belongingness, through all intermediate values Zadeh employed the concept of a membership function assigning to each element a number from the unit interval to indicate the intensity of belongingness. Zadeh further defined basic operations on fuzzy sets as essentially extensions of their conventional ('ordinary') counterparts. Lotdfi Zadeh, Professor in the Graduate School, Computer Science Division Department of Elec. Eng. and Comp Sciences, University of California Berkeley, CA 94720 -1776 Director, Berkeley Initiative in Soft Computing (BISC) http://www.cs.berkeley.edu/People/Faculty/Homepages/zadeh.html In 1995, Dr. Zadeh was awarded the IEEE Medal of Honor "For pioneering development of fuzzy logic and its many diverse applications." In 2001, he received the American Computer Machinery’s 2000 Allen Newell Award for seminal contributions to AI through his development of fuzzy logic. BACKGROUND & DEFINITIONS Zadeh also devised the so-called fuzzy logic: This logic was devised model 'human' reasoning processes comprising: vague predicates: e.g. large, beautiful, small partial truths: e.g. not very true, more or less false linguistic quantifiers: e.g. most, almost all, a few linguistic hedges: e.g. very, more or less. BACKGROUND & DEFINITIONS Charles Elkan, an assistant professor of computer science and engineering at the University of California at San Diego, offers the following definition: "Fuzzy logic is a generalization of standard logic, in which a concept can possess a degree of truth anywhere between 0.0 and 1.0. Standard logic applies only to concepts that are completely true (having degree of truth 1.0) or completely false (having degree of truth 0.0). Fuzzy logic is supposed to be used for reasoning about inherently vague concepts, such as 'tallness.' For example, we might say that 'President Clinton is tall,' with degree of truth of 0.9. The term fuzzy logic is used in two senses: •Narrow sense: Fuzzy logic is a branch of fuzzy set theory, which deals (as logical systems do) with the representation and inference from knowledge. Fuzzy logic, unlike other logical systems, deals with imprecise or uncertain knowledge. In this narrow, and perhaps correct sense, fuzzy logic is just one of the branches of fuzzy set theory. •Broad Sense: fuzzy logic synonymously with fuzzy set theory 2.1 Basic concepts Crisp (Classic、Boolean) Sets universe of discourse characteristic function Crisp Logic Crisp logic is concerned with absolutes-true or false, there is no in-between. Example: Rule: If the temperature is higher than 80F, it is hot; otherwise, it is not hot. Cases: Temperature = 100F Hot Temperature = 80.1F Hot Temperature = 79.9F Not hot Temperature = 50F Not hot Membership function of crisp logic True 1 HOT False 0 80F Temperature If temperature >= 80F, it is hot (1 or true); If temperature < 80F, it is not hot (0 or false). Drawbacks of crisp logic The membership function of crisp logic fails to distinguish between members of the same set. Conception of Fuzzy Logic Many decision-making and problem-solving tasks are too complex to be defined precisely however, people succeed by using imprecise knowledge Fuzzy logic resembles human reasoning in its use of approximate information and uncertainty to generate decisions. Natural Language Consider: Joe is tall -- what is tall? Joe is very tall -- what does this differ from tall? Natural language (like most other activities in life and indeed the universe) is not easily translated into the absolute terms of 0 and 1. “false” “true” Fuzzy Sets • Human reasoning often uses vagueness – Individuals cannot be classified into two groups! (either true or false) • Example: The set of tall men – But… what is tall? – Height is all relative – As a descriptive term, tall is very subjective and relies on the context in which it is used • Even a 5ft7 man can be considered "tall" when he is surrounded by people shorter than he is Fuzzy Logic An approach to uncertainty that combines real values [0…1] and logic operations Fuzzy logic is based on the ideas of fuzzy set theory and fuzzy set membership often found in natural (e.g., spoken) language. Example: “Young” Example: Ann is 28, Bob is 35, Charlie is 23, 0.8 in set “Young” 0.1 in set “Young” 1.0 in set “Young” Unlike statistics and probabilities, the degree is not describing probabilities that the item is in the set, but instead describes to what extent the item is the set. Membership function of fuzzy logic Fuzzy values DOM Degree of Membership Young Middle Old 1 0.5 0 25 40 55 Age Fuzzy values have associated degrees of membership in the set. Crisp set vs. Fuzzy set A traditional crisp set A fuzzy set Crisp set vs. Fuzzy set Membership Functions fuzzy set A on the universe of discourse X can be expressed symbolically as the set of ordered pairs •the membership function of x on the set Α •a mapping of the universe of discourse Χ on the closed interval [0,1] •simply a measure of the degree to which x belongs to the set Α Membership Functions Membership functions represent distributions of possibility rather than probability For instance, the fuzzy set Young expresses the possibility that a given individual be young Membership functions often overlap with each others – A given individual may belong to different fuzzy sets (with different degrees) Example temperature of water [0,100] fuzzy variable - Low Α={Low} ‘+’ represents the union operator Fuzzy variable expressed in terms of phrases that combine linguistic variables, linguistic descriptors and hedges fuzzy conditional statement R : IF S1 THEN S2 or symbolically as: S1 →S2 for example: S : Χ is Α fuzzy conditional statement composite conditional statement for example: 2.2 Fuzzy Algorithms Two or more fuzzy statements can be combined with the OR connective to form a fuzzy algorithm 2. 3 Fuzzy Operators min (for minimum) and max (for maximum) PLC AND and OR functions two elements 2. 3 Fuzzy Operators min (for minimum) and max (for maximum) one element minimum (inf) or maximum (sup) of all the elements of the set two sets 2. 3 Fuzzy Operators discrete sets 2. 4 Operations on Fuzzy Sets null set identical membership function is zero everywhere membership functions are identical everywhere on the universe of discourse 2. 4 Operations on Fuzzy Sets subset union intersection product Operations A AB B AB A 2. 5 Algebraic Properties of Fuzzy Sets 2. 5 Algebraic Properties of Fuzzy Sets The following properties apply only to fuzzy sets: 2. 5 Algebraic Properties of Fuzzy Sets Example 2.6 Linguistic Variables Variables used in fuzzy systems to express qualities such as height, which can take values such as “tall”, “short” or “very tall”. These values define subsets of the universe of discourse. 2.6 Linguistic Variables terms primary terms which are labels of fuzzy sets, such as High, Low, Small, Medium, Zero, negation NOT and connectives AND and OR, hedges such as very, nearly, almost and markers such as parentheses ( ). 2.6 Linguistic Variables primary terms-----continuous or discrete membership functions Gaussian-like membership functions 2.6 Linguistic Variables primary functions terms-----continuous or discrete membership generic S functions 2.6 Linguistic Variables primary terms-----continuous or discrete membership functions Π function 2.6 Linguistic Variables primary terms-----continuous or discrete membership functions constructed from standardized trapezoidal and triangular functions. 2.6 Linguistic Variables primary terms-----continuous or discrete membership functions Discrete fuzzy set --- sets of singletons on a finite universe of discourse example universe of discourse linguistic variables small, medium and large 2.7 Connectives negation NOT and connectives AND and OR Negation (NOT) and the connectives AND and OR can be defined in terms of the complement, union and intersection operations respectively. AND different universes of discourse 2.7 Connectives negation NOT and connectives AND and OR OR NOT operator same universes of discourse negation Hedges A fuzzy set qualifier, such as “very”, “quite”, “extremely”, or “somewhat” When applied to fuzzy set, the new set will be produce Eg: apply “very” to “tall people” – subset of “tall people” will produce that called “very tall people” The meanings of these hedges are fairly subjective usually use a systematic mathematic definition for the hedges so that it can be applied logically. Often a hedges is applied by raising the set’s membership function to an appropriate power 2.7 Connectives hedges Hedge is an intensifier Example: LV = tall, LV1 = very tall, LV2 = somewhat tall ‘very’ operation: μvery tall(x) = μ2tall(x) ‘somewhat’ operation: μsomewhat tall(x) = √(μtall(x)) somewhat tall tall 1 μtall(h) 0 very tall h An Example: Consider a set of numbers: X = {1, 2, ….. 10}. Johnny’s understanding of numbers is limited to 10, when asked he suggested the following. Sitting next to Johnny was a fuzzy logician noting : Comment ‘Degree of membership’ 10 ‘Surely’ 1 9 ‘Surely’ 1 8 ‘Quite poss.’ 0.8 7 ‘Maybe’ 0.5 6 ‘In some cases, not usually’ 0.2 5, 4, 3, 2, 1 ‘Definitely Not’ 0 ‘Large Number’ We can denote Johnny’s notion of ‘large number’ by the fuzzy set A =0/1+0/2+0/3+0/4+0/5+ 0.2/6 + 0.5/7 + 0.8/8 + 1/9 + 1/10 Fuzzy (sub-)sets: Membership Functions For the sake of convenience, usually a fuzzy set is denoted as: A = A(xi)/xi + …………. + A(xn)/xn that belongs to a finite universe of discourse: A ~ {x1, x2 ,.......,xn } where A(xi)/xi (a singleton) is a pair “grade of membership element”. Johnny’s large number set membership function can be denoted as: ‘Large Number’ (.) 10 A (10) = 1 9 A (9) = 1 8 A (8) = 0.8 7 A (7) = 0.5 6 A (6) = 0.2 5, 4, 3, 2, 1 A (5) = A (4) = A (3) = A (2) = A (10) = 0 Johnny’s large number set membership function can be used to define ‘small number’ set B, where B (.)= NOT (A (.) ) = 1 - A (.): ‘Small Number’ (.) 10 B(10) = 0 9 B (9) = 0 8 B (8) = 0.2 7 B (7) = 0.5 6 B (6) = 0.8 5, 4, 3, 2, 1 B (5) = B (4) = B (3) = B (2) = B (10) = 1 Concept of Hedge Johnny’s large number set membership function can be used to define ‘very large number’ set C, where C (.)= A (.)* A (.) Number Very Large (C (.)) 10 1 9 1 8 0.64 7 0.25 6 0.04 5, 4, 3, 2, 1 0

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