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Fuzzy Logic
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FUZZY LOGIC WHAT IS FUZZY LOGIC? In classical set theory members of a given set either have full membership in that set or no membership at all, i.e. the "degree of membership" is restricted to either TRUE or FALSE or 1 and 0. Linguistic variables (central to fuzzy logic manipulations) hold values that are uniformly distributed between 0 and 1, and are dependent on the linguistic term. Applications may be computed in either the fuzzy linguistic domain or the conventional crisp domain. Linguistic variables are relevant in applications involving human interface as tasks commonly done by humans cannot easily be described in crisp terms. Fuzzy logic derives its importance from the fact that most modes of human reasoning and especially common sense reasoning are approximate in nature-they do not have any definite answers and are dependent on many varied factors. In such cases linguistic variables provide a convenient tool to describe a problem. Fuzzy logic invented by Dr. Lotfi Zadeh in the 1960s is used to help computers deal with the approximate. The "linguistic variables" used conveys relative Information about the object under observation and can convey a surprising amount of information. For this reason, it is essential in the design of expert systems, which apply real-world rules to real-world situations. Fuzzy Logic is a very powerful concept which, when applied to control systems allows designers to greatly reduce the number of rules necessary to deploy a desired control strategy. It also allows rules to be crafted in a very linguistic syntax and reduces instabilities and hysteresis. Fuzzy theory and fuzzy systems are used in applications like spellcheckers and palmtop handwriting interpreters-especially in Japan, complicated Kanji strokes can be detected as they're written using fuzzy methods. It is used in controllers for products such as washing machines, video cameras, and ocean drilling systems; and in pattern recognition and predictive software for financial modeling, medical imaging, and manufacturing. The first major commercial application was in a cement kiln control, an operation which requires that an operator, monitor four internal states of the kiln, control four sets of operations, and dynamically manage 40 or 50 "rules of thumb" about their interrelationships, all with the goal of controlling a highly complex set of chemical interactions. FUZZY SET THEORY and TRUTH VALUES: In Boolean logic, conditions are evaluated to be either TRUE or FALSE. There are no intermediate values. Every element either belongs to the set or it does not belong to it. This concept is sufficient for many applications, but we can easily find situations where it lacks in flexibility. In fuzzy logic the possible values range from 0.0 to 1.0 (inclusive), not just 0 and 1. Fuzzy greatly enhances the capability of classical set theory by allowing the "degree of membership" or "truth value" to range over the interval of 0 to 1. Sets in fuzzy logic systems typically describe ranges of operations and are named using linguistic adjectives such as "slow", "medium" or "fast". The degree of membership describes how "slow" or how "fast" a particular value is. Let us consider an example where we define “youthness”. In this case the set S (the universe of discourse) is the set of people. A fuzzy subset YOUNG is also defined, where each person is assigned a degree of membership in the fuzzy subset YOUNG which answers the question "to what degree is person x young?" The membership function is based on the persons age. Eg: young(x) = 1, if age(x) <= 20., =(30-age(x))/10, if 20 < age(x) <= 30, =0, if age(x) > 30 } A graph of this looks like: According to the graph if Rohans age is 10 then his degree of youth is 1.00 and if Krishnas age is 83 his degree of youth is 0. However if Edward and Colin are aged 21 and 26 their degree of youth will be 0.90 and 0.40 respectively. A membership function can be defined as a function that maps each point of fuzzy set A to the real interval [0.0, 1.0] such that as m(A(x)) approaches the grade of membership for x in A increases. LOGICAL OPERATIONS ON FUZZY SETS In boolean logic there are the union (or), intersection (and) and not operators. These operators exist in fuzzy logic too, but are defined differently: A AND B = MIN(m(A(x)), m(B(x))) (minimum operator for intersection of two fuzzy sets) A OR B = MAX(m(A(x)), m(B(x))) (maximum operator for union of two fuzzy sets) A' (NOT A) = 1 - mA(x) These operations are illustrated with the help of graphs: Let A be a fuzzy interval between 5 and 8 and B be a fuzzy number about 4 The following figure shows the fuzzy set between 5 and 8 AND about 4 (blue line). The Fuzzy set between 5 and 8 OR about 4 is given below ( blue line). This figure gives an example for a negation. The blue line is the NEGATION of the fuzzy set A.
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