Book•
Fuzzy Sets and Fuzzy Logic: Theory and Applications
01 Jan 1995-
TL;DR: Fuzzy Sets and Fuzzy Logic is a true magnum opus; it addresses practically every significant topic in the broad expanse of the union of fuzzy set theory and fuzzy logic.
Abstract: Fuzzy Sets and Fuzzy Logic is a true magnum opus. An enlargement of Fuzzy Sets, Uncertainty,
and Information—an earlier work of Professor Klir and Tina Folger—Fuzzy Sets and Fuzzy Logic
addresses practically every significant topic in the broad expanse of the union of fuzzy set theory
and fuzzy logic. To me Fuzzy Sets and Fuzzy Logic is a remarkable achievement; it covers its vast
territory with impeccable authority, deep insight and a meticulous attention to detail.
To view Fuzzy Sets and Fuzzy Logic in a proper perspective, it is necessary to clarify a point
of semantics which relates to the meanings of fuzzy sets and fuzzy logic.
A frequent source of misunderstanding fias to do with the interpretation of fuzzy logic. The
problem is that the term fuzzy logic has two different meanings. More specifically, in a narrow
sense, fuzzy logic, FLn, is a logical system which may be viewed as an extension and generalization
of classical multivalued logics. But in a wider sense, fuzzy logic, FL^ is almost synonymous
with the theory of fuzzy sets. In this context, what is important to recognize is that: (a) FLW is much
broader than FLn and subsumes FLn as one of its branches; (b) the agenda of FLn is very different
from the agendas of classical multivalued logics; and (c) at this juncture, the term fuzzy logic is
usually used in its wide rather than narrow sense, effectively equating fuzzy logic with FLW
In Fuzzy Sets and Fuzzy Logic, fuzzy logic is interpreted in a sense that is close to FLW. However,
to avoid misunderstanding, the title refers to both fuzzy sets and fuzzy logic.
Underlying the organization of Fuzzy Sets and Fuzzy Logic is a fundamental fact, namely,
that any field X and any theory Y can be fuzzified by replacing the concept of a crisp set in X and Y
by that of a fuzzy set. In application to basic fields such as arithmetic, topology, graph theory, probability
theory and logic, fuzzification leads to fuzzy arithmetic, fuzzy topology, fuzzy graph theory,
fuzzy probability theory and FLn. Similarly, hi application to applied fields such as neural network
theory, stability theory, pattern recognition and mathematical programming, fuzzification leads to
fuzzy neural network theory, fuzzy stability theory, fuzzy pattern recognition and fuzzy mathematical
programming. What is gained through fuzzification is greater generality, higher expressive
power, an enhanced ability to model real-world problems and, most importantly, a methodology for
exploiting the tolerance for imprecision—a methodology which serves to achieve tractability,
Citations
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TL;DR: Based on score function and accuracy function, a method is introduced for the comparison between two intuitionistic fuzzy values and some aggregation operators are developed, such as the intuitionism fuzzy weighted averaging operator, intuitionists fuzzy ordered weighted averaging operators, and intuitionistic fuzziness hybrid aggregation operator, for aggregating intuitionist fuzzy values.
Abstract: An intuitionistic fuzzy set, characterized by a membership function and a non-membership function, is a generalization of fuzzy set. In this paper, based on score function and accuracy function, we introduce a method for the comparison between two intuitionistic fuzzy values and then develop some aggregation operators, such as the intuitionistic fuzzy weighted averaging operator, intuitionistic fuzzy ordered weighted averaging operator, and intuitionistic fuzzy hybrid aggregation operator, for aggregating intuitionistic fuzzy values and establish various properties of these operators.
2,131 citations
Additional excerpts
...and is a well-known -norm satisfying [46]....
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...where is a well-known -conorm satisfying [46]....
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Book•
01 Jan 2004TL;DR: In this article, the authors present a set of heuristics for solving problems with probability and statistics, including the Traveling Salesman Problem and the Problem of Who Owns the Zebra.
Abstract: I What Are the Ages of My Three Sons?.- 1 Why Are Some Problems Difficult to Solve?.- II How Important Is a Model?.- 2 Basic Concepts.- III What Are the Prices in 7-11?.- 3 Traditional Methods - Part 1.- IV What Are the Numbers?.- 4 Traditional Methods - Part 2.- V What's the Color of the Bear?.- 5 Escaping Local Optima.- VI How Good Is Your Intuition?.- 6 An Evolutionary Approach.- VII One of These Things Is Not Like the Others.- 7 Designing Evolutionary Algorithms.- VIII What Is the Shortest Way?.- 8 The Traveling Salesman Problem.- IX Who Owns the Zebra?.- 9 Constraint-Handling Techniques.- X Can You Tune to the Problem?.- 10 Tuning the Algorithm to the Problem.- XI Can You Mate in Two Moves?.- 11 Time-Varying Environments and Noise.- XII Day of the Week of January 1st.- 12 Neural Networks.- XIII What Was the Length of the Rope?.- 13 Fuzzy Systems.- XIV Everything Depends on Something Else.- 14 Coevolutionary Systems.- XV Who's Taller?.- 15 Multicriteria Decision-Making.- XVI Do You Like Simple Solutions?.- 16 Hybrid Systems.- 17 Summary.- Appendix A: Probability and Statistics.- A.1 Basic concepts of probability.- A.2 Random variables.- A.2.1 Discrete random variables.- A.2.2 Continuous random variables.- A.3 Descriptive statistics of random variables.- A.4 Limit theorems and inequalities.- A.5 Adding random variables.- A.6 Generating random numbers on a computer.- A.7 Estimation.- A.8 Statistical hypothesis testing.- A.9 Linear regression.- A.10 Summary.- Appendix B: Problems and Projects.- B.1 Trying some practical problems.- B.2 Reporting computational experiments with heuristic methods.- References.
2,089 citations
TL;DR: This paper demonstrates that it is unnecessary to take the route from general T2 FS to IT2 FS, and that all of the results that are needed to implement an IT2 FLS can be obtained using T1 FS mathematics.
Abstract: To date, because of the computational complexity of using a general type-2 fuzzy set (T2 FS) in a T2 fuzzy logic system (FLS), most people only use an interval T2 FS, the result being an interval T2 FLS (IT2 FLS). Unfortunately, there is a heavy educational burden even to using an IT2 FLS. This burden has to do with first having to learn general T2 FS mathematics, and then specializing it to an IT2 FSs. In retrospect, we believe that requiring a person to use T2 FS mathematics represents a barrier to the use of an IT2 FLS. In this paper, we demonstrate that it is unnecessary to take the route from general T2 FS to IT2 FS, and that all of the results that are needed to implement an IT2 FLS can be obtained using T1 FS mathematics. As such, this paper is a novel tutorial that makes an IT2 FLS much more accessible to all readers of this journal. We can now develop an IT2 FLS in a much more straightforward way
1,892 citations
Cites methods from "Fuzzy Sets and Fuzzy Logic: Theory ..."
...Present approaches to doing this use the Extension Principle [24], alpha-cuts, or interval arithmetic (e.g., [ 8 ])....
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01 Nov 1999
TL;DR: Because the fuzzy commitment scheme is tolerant of error, it is capable of protecting biometric data just as conventional cryptographic techniques, like hash functions, are used to protect alphanumeric passwords.
Abstract: We combine well-known techniques from the areas of error-correcting codes and cryptography to achieve a new type of cryptographic primitive that we refer to as a fuzzy commitment scheme. Like a conventional cryptographic commitment scheme, our fuzzy commitment scheme is both concealing and binding: it is infeasible for an attacker to learn the committed value, and also for the committer to decommit a value in more than one way. In a conventional scheme, a commitment must be opened using a unique witness, which acts, essentially, as a decryption key. By contrast, our scheme is fuzzy in the sense that it accepts a witness that is close to the original encrypting witness in a suitable metric, but not necessarily identical.This characteristic of our fuzzy commitment scheme makes it useful for applications such as biometric authentication systems, in which data is subject to random noise. Because the scheme is tolerant of error, it is capable of protecting biometric data just as conventional cryptographic techniques, like hash functions, are used to protect alphanumeric passwords. This addresses a major outstanding problem in the theory of biometric authentication. We prove the security characteristics of our fuzzy commitment scheme relative to the properties of an underlying cryptographic hash function.
1,744 citations
Cites background from "Fuzzy Sets and Fuzzy Logic: Theory ..."
...Uncertainty leads to introduction of fuzzy sets and fuzzy logic[ 2 ] in to the protocol itself....
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TL;DR: The issue of having to choose a best alternative in multicriteria decision making leads the problem of comparing Pythagorean membership grades to be considered, and a variety of aggregation operations are introduced for these Pythagorian fuzzy subsets.
Abstract: We first look at some nonstandard fuzzy sets, intuitionistic, and interval-valued fuzzy sets. We note both these allow a degree of commitment of less then one in assigning membership. We look at the formulation of the negation for these sets and show its expression in terms of the standard complement with respect to the degree of commitment. We then consider the complement operation. We describe its properties and look at alternative definitions of complement operations. We then focus on the Pythagorean complement. Using this complement, we introduce a class of nonstandard Pythagorean fuzzy subsets whose membership grades are pairs, (a, b) satisfying the requirement a
2
+ b
2
≤ 1. We introduce a variety of aggregation operations for these Pythagorean fuzzy subsets. We then look at multicriteria decision making in the case where the criteria satisfaction are expressed using Pythagorean membership grades. The issue of having to choose a best alternative in multicriteria decision making leads us to consider the problem of comparing Pythagorean membership grades.
1,706 citations
Additional excerpts
...We now make some comments about the complement operator, we refer the reader to Klir [12] for more details....
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References
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Book•
01 Feb 2006
2,467 citations
"Fuzzy Sets and Fuzzy Logic: Theory ..." refers background in this paper
...( x - 1 when* 6 [1,2] 3 - x when x 6 [2, 3] 0 otherwise....
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...[2,2] [2,2] [2,2] [1,2] [1,2] [1,2] [1,3] [1,3] [1,4] [0,4]...
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...[2,5] - [1,3] = [ -1 , 4] [0,1] - [-6, 5] = [-5, 7],...
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...For example, the set A = [0, 2] U [3, 5] is not convex, as can be shown by producing one of an infinite number of possible counter-examples: let r = 1, s = 4, and X = 0....
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...Calculate the following: (a) [-1,2]+ [1,3]; (b) [-2,4]-[3,6]; (c) [-3, 4]- [-3, 4]; (d) [-4,6]/[l,2]....
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01 Mar 1986
TL;DR: Fuzzy set theory has a number of properties that make it suitable for formalizing the uncertain information upon which medical diagnosis and treatment is usually based, and trials performed with the medical expert system CADIAG-2 suggest that it might be a suitable basis for the development of a computerized diagnosis system.
Abstract: Fuzzy set theory has a number of properties that make it suitable for formalizing the uncertain information upon which medical diagnosis and treatment is usually based. Firstly, it defines inexact medical entities as fuzzy sets. Secondly, it provides a linguistic approach with an excellent approximation to texts. Finally, fuzzy logic offers reasoning methods capable of drawing approximate inferences. These facts suggest that fuzzy set theory might be a suitable basis for the development of a computerized diagnosis system. This is verified by trials performed with the medical expert system CADIAG-2, which uses fuzzy set theory to formalize medical relationships and fuzzy logic to model the diagnostic process.
392 citations
TL;DR: In this paper, it was proved that distributivity, monotonicity and boundary conditions are essential assumptions for truth-functional logical connectives for fuzzy sets, under reasonable hypotheses (especially distributivity).
334 citations
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TL;DR: The decision trees method is extended to the case when the involved data appear as words belonging to the common language whose semantic representations are fuzzy sets, and a reformalization of the basic concepts of probability and utility theory is carried out.
293 citations
Additional excerpts
...[2, 5] + [1, 3] = [3, 8] [0,1] + [~6,5] = [-6, 6],...
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...[6,6] [5,6] [4,6] P,6] [2,6] [2,7] [2,8] [2,8] [1,9] [0,10] "X...
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01 Sep 1991
TL;DR: A complete design procedure for a fuzzy three-term PID controller containing the rules along with the quantization and tuning procedures by means of input and output mapping factors is presented.
Abstract: A complete design procedure for a fuzzy three-term PID controller is presented. A plant model is not required to achieve this design. A reduced look-up table containing the rules along with the quantization and tuning procedures by means of input and output mapping factors are introduced. The scaling factors of the output error time sequences are preselected arbitrarily and the search for an optimal input-to-output mapping factors ratio is performed through a phase diagram analysis. the applicability of the design procedure is demonstrated through computer simulations. >
98 citations
"Fuzzy Sets and Fuzzy Logic: Theory ..." refers background in this paper
...( x - 1 when* 6 [1,2] 3 - x when x 6 [2, 3] 0 otherwise....
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...To overview them, let A = [ai, o j , B = [bu b2], C = [a, c2], 0 = [0,0], 1 = [1,1]....
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...[2,5] - [1,3] = [ -1 , 4] [0,1] - [-6, 5] = [-5, 7],...
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...2 are also similar in the sense that numbers outside the interval [1, 3] are virtually excluded from the associated fuzzy sets, since their membership grades are either equal to 0 or negligible....
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...To show that distributivity does not hold in general, let A = [0,1], B = [1,2], C = [-2, -1] , Then, A • fl = [0, 2], A • C = [-2, 0], B + C = [-1,1], and...
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