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Notes on Linguistic variables from extracts of Zadeh What are linguistic variables? What is a fuzzy algorithm? What is principle of incompatibility? How is human intelligence different from machine intelligence? Linguistic and fuzzy variables In quantitative approach if we have a conditional statement like “if x is 6 then y is 10”. The same statement in fuzzy logic approach can be given by Basic operations of fuzzy sets(RECAP) Example of these basic operators Concentration and dilation of linguistic variables CON(A) and DIL(A) are conventionally used to represent hedges very and more or less. combination of NOT(A), A AND B, A OR B, and CON(A) and DIL(A) can be used in the construction of composite linguistic term. The values of a linguistic variable are labels of fuzzy subsets of U which have the form of phrases or sentences in a natural or artificial language. If U is the collection of integers and age is a linguistic variable labelled x, then the values of x might be young, not young, very young, not very young, old and not old, not very old, not young and not old. x=hu where h is a hedge u is a term with a specified meaning. x = very tall man, Hedges serve the function of generating a larger set of values for a linguistic variable from a small collection of primary terms. Hedge acts as an intensifier generating a subset of the set on which it operates. As an example or Another example of hedges in a linguistic variable small COMPUTATION OF THE MEANING OF VALUES OF A LINGUISTIC VARIABLE In addition to hu the terms not, or and and can also be used for the computation of linguistic variables where small is defined as with the universe of discourse being A linguistic variable is characterized by a quintuple(x,T(x), X,G,M) x name of the variable T(x) term set of x(set of linguistic variables or terms) X universe of discourse G syntactic rule which generates the terms in T(x) Msemantic rule which associates with each linguistic value A its meaning M(A) M(A) fuzzy set in X. Example If age is interpreted as a linguistic variable then its term set T(age) T(age) = { young, not young, very young, not very young,…., Middle aged, not middle aged…………. Old, not old, very old, not very old, more or less old,….. Not very young and not very old,……………….} X=[0,100] -- age of persons in the span of 0 and 100 years “age is young” linguistic variable age is assigned to the linguistic value “young” Age=20 assignment of numerical value 20 to the numerical variable age Syntactic rule- refers to the way the linguistic values in the term set T(age) are generated Semantic rule- Defines the membership function of each linguistic value of the term set Primary terms(young, middle aged, old) Modified by not, and, or Hedges(very, more or less, quite, extremely ) Connectives – and, or, either and neither. Try out these µyoung(x) = bell(x,20,2,0) = [1+(x/40)4] -1 µold(x) = bell(x,30,3,100) = [1+(x-100/30)6] -1 Find more or less old not young and not old young but not too young extremely old Example of contrast intensifier follow my worked out example in my notes Orthogonality A term set T=t1,t2, ….tn of a linguistic variable x on the universe X is orthogonal if it fulfils the following property n ( x) 1, x X ti i 1 Where the ti’s are convex and normal fuzzy sets defined on X and these fuzzy sets make up the term set T. Fuzzy If-Then Rules • Fuzzy rule, fuzzy implication, fuzzy conditional statement General format: If x is A then y is B A B x is A : antecedent (or premise) y is B : consequence (or conclusion) Examples: • If pressure is high, then volume is small. If the • road is slippery, then driving is dangerous If a tomato is red, then it is ripe • speed If the road is slippery, If the is high, then apply the brake a little then driving is dangerous. What do u mean by “if x is A then y is B” • Ifdescribes a tomato red,twothen itx is The expression a relationis between variables and ripe. y, which suggests that a fuzzy if-then rule be defined as a binary fuzzy relation R on the If the speed is high, then apply the product of•X*Y How can :If x brake is A then Ya is B” be interpreted? little. A If-then rule is a relation between 2 variables x and y; define as a binary fuzzy relation R on X Y. Two ways to interpret “If x is A then y is B”: y A coupled with B B y A entails B B x x Incomplete------------------------------------------------------------------------------------------ The inference of the form Where A,A’,B,B’ are fuzzy concepts is called as fuzzy conditional inference. As an example If A=A’and A and B are non-fuzzy then the above equation reduces to classical modus ponens of a two valued logic. In classical logic, modus ponendo ponens (Latin for the way that affirms by affirming;[1] often abbreviated to MP or modus ponens) is a valid, simple argument form sometimes referred to as affirming the antecedent or the law of detachment. Modus ponens is a very common rule of inference, and takes the following form: If P, then Q. P. Therefore, Q But if A,A’ and B are fuzzy and A!=A’ then this form of inference cannot be made by classical modus ponens. For this Zadeh formalized an inference rule named as “compositional rule of inference”.