Nguyen, H.T. and Walker, E.A. (1999) A First Course in Fuzzy Logic, CRC Press, USA.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....some other modifier m . So, from mathematical viewpoint, we get a mapping that maps each modifier m into some other modifier m . There are two ways to describe this mapping: ffl We can describe this mapping indirectly, i.e. we can use the known formulas for fuzzy implication (see, e.g. [2, 3, 4]) and find the mapping that these formulas lead to. For several known implication operations, such mappings were (partially) described in [1] ffl We can also try to describe this mapping directly. In other words, we can try to describe fuzzy modus ponens as a calculus of logical modifiers. This ....

H. T. Nguyen and E. A. Walker, A First Course in Fuzzy Logic, CRC Press, Boca Raton, Florida, 1996 (to appear). 9

....interval [x i Gamma ; x i ] we have a nested family of intervals [x i Gamma j ; x i j ] corresponding to different degrees of certainty. Such a nested family of intervals is also called a fuzzy set, because it turns out to be equivalent to a more traditional definition of fuzzy set [2, 12, 14, 15] (if a traditional fuzzy set is given, then different intervals from the nested family can be viewed as ff cuts corresponding to different levels of uncertainty ff) In these terms, in addition to detecting and deleting duplicates under interval uncertainty, we must also detect and delete them ....

....3 , ffl etc. In other words, to solve a fuzzy problem, we solve several interval problems corresponding to different levels of uncertainty. It is worth mentioning that this interval approach to solving a fuzzy problem is in line with many other algorithms for processing fuzzy data; see, e.g. [2, 12, 14, 15]. Acknowledgments This work was supported in part by NASA grants NCC5209 and NCC2 1232, by the Air Force Office of Scientific Research grant F49620 00 1 0365, and by NSF grants CDA 9522207, EAR 0112968, EAR 0225670, and 9710940 Mexico Conacyt. This research was partly done when V.K. was a ....

H. T. Nguyen and E. A. Walker, First Course in Fuzzy Logic, CRC Press, Boca Raton, FL, 1999.

.... most probable , etc. In other words, instead of a two valued logic, we have a multi valued logic, in which instead of a truth value that can take only two possible values, we have degree of belief that has many possible values. The most widely used multi valued logic is fuzzy logic (see, e.g. [8, 11]) in which degrees of belief are described by numbers from the interval [0; 1] 1 corresponds to absolutely true , 0 corresponds to absolutely false , and values in between 0 and 1 describe intermediate degrees of belief. 1.2 The basic operators: As with traditional two valued logic, ....

H.T. Nguyen and E.A. Walker, A First Course in Fuzzy Logic, CRC Press, Boca Raton, Florida, 1997.

.... to make inferences: If x is P then y is Q I(P (x) Q (y) x is P # y is Q # Q # (y) Sup T ( P # (x) I(P (x) Q (y) If we want to use the rules and its associated implications to make inferences, the T conditional associated to the implication must fulfil next inequality [2], for some t norm T: T ( P (x) I(P (x) Q (y) Q (y) that is, T (a, I(a, b) b for all a, b Theorem 2.1 I T is a T conditional for any t norm T Proof T 1 (a, I T (a, b) T (#(a) #(b) #(b) b. That is to say, I T is a T conditional. # Let s only consider the case ....

Hung T. Nguyen and Elbert A. Walker. A First course in Fuzzy Logic. Chapman & Hall/Crc, 2000.

.... error component, which correspond to interval uncertainty; see, e.g. 2, 4, 6] ffl in addition to these bounds in which experts are absolutely sure, experts have smaller bounds with reasonable but not absolute certainty; these bounds are naturally described by fuzzy techniques; see, e.g. [1, 5, 10]. 2. Case Study: Monoclinic Transverse Isotropic Material Let us first recall how the dispersion curve depends on the elastic constants; this dependence is described, e.g. in [8] see also [7] In elastic materials, usually, stress oe ij is linearly related to strain e kl , i.e. oe ....

....is possible. In other words, instead of a (crisp) set of possible combinations, we get a fuzzy set. To be more precise, we get a nested family of sets corresponding to different levels of certainty; such a nested family of sets is indeed equivalent to a more traditional definition of fuzzy set [1, 5, 10]: if a traditional fuzzy set is given, then different sets from the nested family can be viewed as ff cuts corresponding to different levels of uncertainty ff. 6. Solution: Probabilistic and Interval Cases What can we reconstruct An experimental analysis. Before designing algorithms for solving ....

H. T. Nguyen and E. A. Walker, First Course in Fuzzy Logic, CRC Press, Boca Raton, FL, 1999.

....by ) which is defined by: a 1 a; here indicates the usual subtraction of real numbers. Hence (L, A, V, is a de Morgan lattice, i.e. a bounded lattice, distributive with respect to the A, V operations, order inverting with respect to the operation and satisfying de Morgan s laws [19]. In addition, L, is a complete lattice and totally ordered. We will be concerned with fuzzy (sub)sets of U; fuzzy sets are identified with their membership functions, which take values in L. In other words, a fuzzy set is a function A: U L. Fuzzy sets will be denoted by uppercase letters: ....

H.T. Ngyen and E.A. Walker. A First Course in Fuzzy Logic. CRC Press 1997.

....cuts are closed intervals of a reference lattice (X, It appears that fuzzy intervals (in this lattice theoretic sense) have not been studied previously. A special case which has been extensively studied is that of fuzzy intervals with the reference lattice (X, being a set of real numbers [7]. Some connections between this special case and the more general case studied here will be discussed briefly in Section 5. As mentioned, our study of fuzzy intervals is lattice theoretic. We establish some basic properties of fuzzy intervals and we show the following: given a complete lattice ....

....V, A will be used without danger of confusion. Definition 2.2 For M, N F we write M N if[for all x X we have: M(x) N(x) Definition 2. 3 For M, N F: we define the fuzzy set M V N by: M V N) x) M(x) V N(x) we define the fuzzy set M A N by: M A N) x) M(x) A N(x) It is well known [7] that is an order on F and that (F, 5, V, A) is a complete and distributive lattice with sup(M, N) M V N, inf(M, N) M A N. Definition 2.4 Given a fuzzy set M : X L, the p cut of M is denoted by Mp and defined by x: M(x) p . We will need some properties of p cuts, summarized in the ....

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H.T. Nguyen and E.A. Walker. A First Course on Fuzzy Logic. CRC Press, Boca Raton, 1997.

....get the desired explanation (at least for a; b 0) of the implication operation f (a; b) b . In two other cases, we get new operations that are worth trying. 4 2. f (a; b) a Delta b is a particular case of a continuous strictly Archimedean Gammaoperation (t Gammanorm) see, e.g. [7,10]) A generic case is f (a; b) a) Delta (b) for some continuous strictly monotonic function : 0; 1] 0; 1] For this case, the above described sets of axioms lead to f (a; b) f ( a) b) where f is an implication operation described in the Main Result. 3. A ....

H. T. Nguyen and E. A. Walker, A First Course in Fuzzy Logic (CRC Press, Boca Raton, Florida, 1997).

....be written as [a, b] for any a, b such that a ; b. crisp) hyperoperation is a mapping o :XxX P(X) a L fuzzy hyperoperation is a mapping : XxX F(X) 5. Given a L fuzzy set M: X X, the p cut of M is denoted by Mp and defined by Mp x: M(x) p . For some basic properties of p cuts see [13, 8]. Two particularly important facts are (for details see [8] a) a fuzzy set is uniquely determined by its p cuts; b) a family of sets fp which has certain properties ( p cut properties ) can be used to pox define a fuzzy set M in a manner such that for every p X we have Mp = Mp. 6. For ....

....For all a, b X we define the L fszz set a Y b b defining for eer x X: T ) a v : v ; 4) we define the L fszz set a A b b defining for eer x X: Proposition 3.23 For all a, b X and p X we hae Proof. Follows from the construction of a Y b, a A b as given in Definition 3. 22 (for details see [13]) Definition 3.24 We say M: X X is a L fuzzy interval of (X, iff Vp X: Mp is a closed interval of (X, Definition 3.25 We denote the collection of L fuzzy intervals of X by (X) Proposition 3.26 For all a, b X , the L fuzzy sets a b and a A b are L fuzzy intervals. Proof. As already ....

H.T. Nguyen and E.A. Walker. A First Course on Fuzzy Logic, CRC Press, Boca Raton, 1997.

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Nguyen H. T., and Walker E. A. (1999), First Course in Fuzzy Logic, CRC Press, Boca Raton, Florida.

....minimum enables to use only N = D=ffi) n=2 calls. Thus, quantum algorithms can double the dimension of the problem for which we are able to compute the desired uncertainty. 5 Quantum Algorithms for Fuzzy Computations Formulation of the problem. In fuzzy data processing (see, e.g. [11, 16]) the main objective is: given: ffl fuzzy numbers X 1 ; Xn characterizing our uncertainty about the inputs x 1 ; xn , and ffl the data processing algorithm f(x 1 ; xn ) that transforms compute the fuzzy number Y = f(X 1 ; Xn ) that describes the resulting ....

H. T. Nguyen and E. A. Walker, First Course in Fuzzy Logic, CRC Press, Boca Raton, FL, 1999.

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Nguyen, H.T. and Walker, E.A. (1999) A First Course in Fuzzy Logic, CRC Press, USA.

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Nguyen, H. T. and E. A. Walker: 2000, A First Course in Fuzzy Logic. Boca Raton: Chapman & Hall/CRC, second edition.

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Hung T. Nguyen and Elbert A. Walker. A First Course in Fuzzy Logic. CRC Press, 1997.

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H. T. Nguyen and E. A. Walker. A First Course in Fuzzy Logic. Chapman & Hall/Crc, Boca Raton, 2000.

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H. T. Nguyen and E. Walker, First Course in Fuzzy Logic, CRC Press, Boca Raton, Florida, 1999.

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H.T. Nguyen and E.A. Walker. A First Course on Fuzzy Logic, CRC Press, Boca Raton, 1997. 4

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Nguyen HT, Walker EA (1999) First Course in Fuzzy Logic, CRC Press, Boca Raton, Florida

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H.T. Nguyen and E.A. Walker. A First Course on Fuzzy Logic, CRC Press, Boca Raton, 1997.

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H.T. Nguyen and E.A. Walker. A First Course in Fuzzy Logic, 2nd Edition, Chapman and Hall, 1999.

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Nguyen, H.T., and Walker, E.A., 1999. First Course in Fuzzy Logic, CRC Press, Boca Raton, FL.

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H.T. Nguyen and E.A. Walker. A First Course on Fuzzy Logic, CRC Press, Boca Raton, 1997.

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H. T. Nguyen and E. Walker. A First Course in Fuzzy Logic. CRC Press, Boca Raton, 1997.

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Nguyen H.T, Walker E.A. A First Course in Fuzzy Logic, CRC, Boca Raton, 2000. 4

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H. T. Nguyen and E. A. Walker, First Course in Fuzzy Logic, CRC Press, Boca Raton, FL, 1999.

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