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Soft Computing Explains Heuristic Numerical Methods in Data Processing and in Logic Programming
 in Data Processing and in Logic Programming
, 1997
"... We show that fuzzy logic and other soft computing approaches explain and justify heuristic numerical methods used in data processing and in logic programming, in particular, Mmethods in robust statistics, regularization techniques, metric fixed point theorems, etc. Introduction What is soft compu ..."
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Cited by 18 (16 self)
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We show that fuzzy logic and other soft computing approaches explain and justify heuristic numerical methods used in data processing and in logic programming, in particular, Mmethods in robust statistics, regularization techniques, metric fixed point theorems, etc. Introduction What is soft computing good for? Traditional viewpoint. When are soft computing methods (fuzzy, neural, etc.) mostly used now? Let us take, as an example, control, which is one of the major success stories of soft computing (especially of fuzzy methods; see, e.g., (Klir 1995)). ffl In control, if we know the exact equations that describe the controlled system, and if we know the exact objective function of the control, then we can often apply the optimal control techniques developed in traditional (crisp) control theory and compute the optimal control. Even in these situations, we can, in principle, use soft computing methods instead: e.g., we can use simpler fuzzy control rules instead of (more complicated...
On Hardware Support For Interval Computations And For Soft Computing: Theorems
, 1994
"... This paper provides a rationale for providing hardware supported functions of more than two variables for processing incomplete knowledge and fuzzy knowledge. The result is in contrast to Kolmogorov's theorem in numerical (nonfuzzy) case. ..."
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Cited by 9 (5 self)
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This paper provides a rationale for providing hardware supported functions of more than two variables for processing incomplete knowledge and fuzzy knowledge. The result is in contrast to Kolmogorov's theorem in numerical (nonfuzzy) case.
Kolmogorov Complexity and Chaotic Phenomena
 International Journal of Engineering Science
, 2002
"... Born about three decades ago, Kolmogorov Complexity Theory (KC) led to important discoveries that, in particular, give a new understanding of the fundamental problem: interrelations between classical continuum mathematics and reality (physics, biology, engineering sciences, . . . ). ..."
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Cited by 8 (7 self)
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Born about three decades ago, Kolmogorov Complexity Theory (KC) led to important discoveries that, in particular, give a new understanding of the fundamental problem: interrelations between classical continuum mathematics and reality (physics, biology, engineering sciences, . . . ).
Justification of Heuristic Methods in Data Processing Using Fuzzy Theory, with Applications to Detection of Business Cycles From Fuzzy Data
 Proceedings of the 8th IEEE International Conference on Fuzzy Systems FUZZIEEE'99, Seoul, Korea
, 1999
"... In this paper, we show that fuzzy theory can explain heuristic methods in inverse problems and numerical computations. As an example of application of these results, we analyze the problem of detecting business cycles from fuzzy data. 1 Inverse Problems Inverse problems: brief reminder. One of the m ..."
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Cited by 3 (3 self)
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In this paper, we show that fuzzy theory can explain heuristic methods in inverse problems and numerical computations. As an example of application of these results, we analyze the problem of detecting business cycles from fuzzy data. 1 Inverse Problems Inverse problems: brief reminder. One of the main problems of data processing is the inverse problem: we observe the signal y after it has passed through some medium, and we want to reconstruct the original signal x (i.e., we want to reconstruct the original image x from the results y of astronomical measurements). When the original signal is strong enough, i.e., when the signaltonoise ratio is high, the signal arrives practically unchanged (y ≈ x), so reconstruction is rather easy (we can even take the observed signal y as a reasonably good approximation to original signal x). The inverse problem becomes complex when the original signal x is weak, i.e., when the signaltonoise ratio is small. In this case, we can expand the dependence of y on x into a Taylor series and neglect quadratic and higher order terms in this expansion. Thus, we have a linear dependence y = Ax + b (for some matrix A and vector b), and we get a linear equation that we have to solve in order to reconstruct x:
Justi cation of Heuristic Methods in Data Processing Using Fuzzy Theory, with Applications to Detection of Business Cycles From Fuzzy Data", EastWest
 Journal of Mathematics, 1999
"... In this paper, we show that fuzzy theory can explain heuristic methods in inverse problems and numerical computations. As an example of application of these results, we analyze the problem of detecting business cycles from fuzzy data. 1 Inverse Problems Inverse problems: brief reminder. One of the m ..."
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Cited by 1 (1 self)
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In this paper, we show that fuzzy theory can explain heuristic methods in inverse problems and numerical computations. As an example of application of these results, we analyze the problem of detecting business cycles from fuzzy data. 1 Inverse Problems Inverse problems: brief reminder. One of the main problems of data processing is the inverse problem: we observe the signal y after it has passed through some medium, and we want to reconstruct the original signal x (i.e., we want to reconstruct the original image x from the results y of astronomical measurements).
Formalizing the Informal, Precisiating the Imprecise: How Fuzzy Logic Can Help Mathematicians and Physicists by Formalizing Their Intuitive Ideas
"... Abstract. Fuzzy methodology transforms expert ideas { formulated in terms of words from natural language { into precise rules and formulas. In this paper, we show that by applying this methodology to intuitive physical and mathematical ideas, we can get known fundamental physical equations and kn ..."
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Cited by 1 (1 self)
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Abstract. Fuzzy methodology transforms expert ideas { formulated in terms of words from natural language { into precise rules and formulas. In this paper, we show that by applying this methodology to intuitive physical and mathematical ideas, we can get known fundamental physical equations and known mathematical techniques for solving these equations. This fact makes us condent that in the future, fuzzy techniques will help physicists and mathematicians to transform their imprecise ideas into into new physical equations and new techniques for solving these equations. 1 Fuzzy and Physics: Past and Present Fuzzy methodology: main objective. Fuzzy methodology has been invented to transform expert ideas – formulated in terms of words from natural language – into precise rules and formulas, rules and formulas understandable by a computer (and implementable on a computer); see, e.g., [6, 10, 12]. Fuzzy methodology: numerous successes. Fuzzy methodology has led to
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"... Abstract In this paper, we show that fuzzy theory can explain heuristic methods in inverse problems and numerical computations. As an example of application of these results, we analyze the problem of detecting business cycles from fuzzy data. 1 Inverse Problems Inverse problems: brief reminder. One ..."
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Abstract In this paper, we show that fuzzy theory can explain heuristic methods in inverse problems and numerical computations. As an example of application of these results, we analyze the problem of detecting business cycles from fuzzy data. 1 Inverse Problems Inverse problems: brief reminder. One of the main problems of data processing is the inverse problem: we observe the signal y after it has passed through some medium, and we want to reconstruct the original signal x (i.e., we want to reconstruct the original image x from the results y of astronomical measurements). When the original signal is strong enough, i.e., when the signaltonoise ratio is high, the signal arrives practically unchanged (y ss x), so reconstruction is rather easy (we can even take the observed signal y as a reasonably good approximation to original signal x). The inverse problem becomes complex when the original signal x is weak, i.e., when the signaltonoise ratio is small. In this case, we can expand the dependence of y on x into a Taylor series and neglect quadratic and higher order terms in this expansion. Thus, we have a linear dependence y = Ax + b (for some matrix A and vector b), and we get a linear equation that we have to solve in order to reconstruct x: Ax = c; where c = y \Gamma b, y is measured, A and b are known, and x has to be determined. For example, when we reconstruct how a single x depends on time x = x(t), we get a linear integral equation Z A(t; t 0
On The Formulation Of Optimization Under Elastic Contraints (with Control In Mind)
, 1994
"... We give a basic survey of various approaches to defining the maximum point of a (crisp) numerical function over a fuzzy set. This survey is based on several unifying ideas, and includes original comparison results. Motivations and applications will be drawn mainly from control. Keywords: Applicatio ..."
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We give a basic survey of various approaches to defining the maximum point of a (crisp) numerical function over a fuzzy set. This survey is based on several unifying ideas, and includes original comparison results. Motivations and applications will be drawn mainly from control. Keywords: Application and Modelling: Decision Making, Control Theory, Mathematical Programming. Contents Abstract 1.
Fuzzy Justification of Heuristic Methods in Inverse Problems and in Numerical Computations, with Applications to Detection of Business Cycles From Fuzzy and Intuitionistic Fuzzy Data
, 1998
"... and the dependence of y on x into a Taylor series and neglect quadratic and higher order terms in this expansion. Thus, we have a linear dependence y = Ax + b (for some matrix A and vector b), and we get a linear equation that we have to solve in order to reconstruct x: Ax = c; where c = y \Gamma ..."
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and the dependence of y on x into a Taylor series and neglect quadratic and higher order terms in this expansion. Thus, we have a linear dependence y = Ax + b (for some matrix A and vector b), and we get a linear equation that we have to solve in order to reconstruct x: Ax = c; where c = y \Gamma b, y is measured, A and b are known, and x has to be determined. For example, when we reconstruct how a single x depends on time x = x(t), we get a linear integral equation Z A(t; t 0 ) \Delta x(t 0 ) dt =<F
Integrating Domain Knowledge With Data: From Crisp To Probabilistic and Fuzzy Knowledge
, 2000
"... It is well known that prior knowledge about the domain can improve (often drastically) the accuracy of the estimates of the physical quantities in comparison with the estimates which are solely based on the measurement results. In this paper, we show how a known method of integrating crisp dom ..."
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It is well known that prior knowledge about the domain can improve (often drastically) the accuracy of the estimates of the physical quantities in comparison with the estimates which are solely based on the measurement results. In this paper, we show how a known method of integrating crisp domain knowledge with data can be (naturally) extended to the case when the domain knowledge is described in statistical or fuzzy terms. 1 INTEGRATING DOMAIN KNOWLEDGE WITH DATA: FORMULATION OF THE PROBLEM A large part of information about the world comes from measurements. However, for many complex systems, some characteristics are very difficult to measure: e.g., for a jet engine, it is difficult to measure the temperature and pressure inside the jet chamber, where the temperatures are very high; for a human body, it is difficult to measure the characteristics of the internal organs, etc. In many such cases, experts have some knowledge about the domain. It is therefore desirable to us...