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Full-Text Articles in Computer Engineering
A New Look At Fuzzy Theory Via Chu Spaces, Hung T. Nguyen, Berlin Wu, Vladik Kreinovich
A New Look At Fuzzy Theory Via Chu Spaces, Hung T. Nguyen, Berlin Wu, Vladik Kreinovich
Departmental Technical Reports (CS)
We propose to use Chu categories as a general framework for uncertainty analysis, with a special attention to fuzzy theory. We emphasize the fact that by viewing fuzzy concepts as Chu spaces, we can discover new aggregation operators, and model interactions and relationship between fuzzy data; these possibilities are due, in essence, to the category structure of Chu spaces, and especially to their morphisms. This paper is a tutorial introduction to the subject.
An Optimality Criterion For Arithmetic Of Complex Sets, Vladik Kreinovich, Juergen Wolff Von Gudenberg
An Optimality Criterion For Arithmetic Of Complex Sets, Vladik Kreinovich, Juergen Wolff Von Gudenberg
Departmental Technical Reports (CS)
Uncertainty of measuring complex-valued physical quantities can be described by complex sets. These sets can have complicated shapes, so we would like to find a good approximating family of sets. Which approximating family is the best? We reduce the corresponding optimization problem to a geometric one: namely, we prove that, under some reasonable conditions, an optimal family must be shift-, rotation- and scale-invariant. We then use this geometric reduction to conclude that the best approximating low-dimensional families consist of sets with linear or circular boundaries. This result is consistent with the fact that such sets have indeed been successful in …
Chu Spaces: Towards New Foundations For Fuzzy Logic And Fuzzy Control, With Applications To Information Flow On The World Wide Web, Hung T. Nguyen, Vladik Kreinovich, Guoqing Liu
Chu Spaces: Towards New Foundations For Fuzzy Logic And Fuzzy Control, With Applications To Information Flow On The World Wide Web, Hung T. Nguyen, Vladik Kreinovich, Guoqing Liu
Departmental Technical Reports (CS)
We show that Chu spaces, a new formalism used to describe parallelism and information flow, provide uniform explanations for different choices of fuzzy methodology, such as choices of fuzzy logical operations, of membership functions, of defuzzification, etc.
On Average Bit Complexity Of Interval Arithmetic, Chadi Hamzo, Vladik Kreinovich
On Average Bit Complexity Of Interval Arithmetic, Chadi Hamzo, Vladik Kreinovich
Departmental Technical Reports (CS)
In many practical situations, we know only the intervals which contain the actual (unknown) values of physical quantities. If we know the intervals [x] for a quantity x and [y] for another quantity y, then, for every arithmetic operation *, the set of possible values of x*y also forms an interval; the operations leading from [x] and [y] to this new interval are called interval arithmetic operations. For addition and subtraction, corresponding interval operations consist of two corresponding operations with real numbers, so there is no hope of making them faster. The best known algorithms for interval multiplication consists of …
Fuzzy Systems Are Universal Approximators For A Smooth Function And Its Derivatives, Vladik Kreinovich, Hung T. Nguyen, Yeung Yam
Fuzzy Systems Are Universal Approximators For A Smooth Function And Its Derivatives, Vladik Kreinovich, Hung T. Nguyen, Yeung Yam
Departmental Technical Reports (CS)
One of the reasons why fuzzy methodology is successful is that fuzzy systems are universal approximators, i.e., that we can approximate an arbitrary continuous function within any given accuracy by a fuzzy system. In some practical applications (e.g., in control), it is desirable to approximate not only the original function, but also its derivatives (so that, e.g., a fuzzy control approximating a smooth control will also be smooth). In our paper, we show that for any given accuracy, we can approximate an arbitrary smooth function by a fuzzy systems so that not only the function is approximated within this accuracy, …
Why Clustering In Function Approximation? Theoretical Explanation, Vladik Kreinovich, Yeung Yam
Why Clustering In Function Approximation? Theoretical Explanation, Vladik Kreinovich, Yeung Yam
Departmental Technical Reports (CS)
Function approximation is a very important practical problem: in many practical applications, we know the exact form of the functional dependence y=f(x1,...,xn) between physical quantities, but this exact dependence is complicated, so we need a lot of computer space to store it, and a lot of time to process it, i.e., to predict y from the given xi. It is therefore necessary to find a simpler approximate expression g(x1,...,xn) for this same dependence. This problem has been analyzed in numerical mathematics for several centuries, and it is, therefore, one of the most thoroughly analyzed problems of applied mathematics. There are …
Beyond [0,1] To Intervals And Further: Do We Need All New Fuzzy Values?, Yeung Yam, Masao Mukaidono, Vladik Kreinovich
Beyond [0,1] To Intervals And Further: Do We Need All New Fuzzy Values?, Yeung Yam, Masao Mukaidono, Vladik Kreinovich
Departmental Technical Reports (CS)
In many practical applications of fuzzy methodology, it is desirable to go beyond the interval [0,1] and to consider more general fuzzy values: e.g., intervals, or real numbers outside the interval [0,1]. When we increase the set of possible fuzzy values, we thus increase the number of bits necessary to store each degree, and therefore, increase the computation time which is needed to process these degrees. Since in many applications, it is crucial to get the result on time, it is therefore desirable to make the smallest possible increase. In this paper, we describe such smallest possible increases.
Time-Bounded Kolmogorov Complexity May Help In Search For Extra Terrestrial Intelligence (Seti), Martin Schmidt
Time-Bounded Kolmogorov Complexity May Help In Search For Extra Terrestrial Intelligence (Seti), Martin Schmidt
Departmental Technical Reports (CS)
One of the main strategies in Search for Extra Terrestrial Intelligence (SETI) is trying to overhear communications between advanced civilizations. However, there is a (seeming) problem with this approach: advanced civilizations, most probably, save communication expenses by maximally compressing their messages, and the notion of a maximally compressed message is naturally formalized as a message x for which Kolmogorov complexity C(x) is close to its length l(x), i.e., as a "random" message. In other words, a maximally compressed message is indistinguishable from the truly random noise, and thus, trying to detect such a message does not seem to be a …
Intervals Is All We Need: An Argument, Masao Mukaidono, Yeung Yam, Vladik Kreinovich
Intervals Is All We Need: An Argument, Masao Mukaidono, Yeung Yam, Vladik Kreinovich
Departmental Technical Reports (CS)
In many practical applications of fuzzy methodology, it is desirable to go beyond the interval [0,1] and to consider more general fuzzy values: e.g., intervals, or more general sets of values. In this paper, we show that under some reasonable assumptions, there is no need to go beyond intervals.
Extending T-Norms Beyond [0,1]: Relevant Results Of Semigroup Theory, Yeung Yam, Vladik Kreinovich
Extending T-Norms Beyond [0,1]: Relevant Results Of Semigroup Theory, Yeung Yam, Vladik Kreinovich
Departmental Technical Reports (CS)
Originally, fuzzy logic was proposed to describe human reasoning. Lately, it turned out that fuzzy logic is also a convenient approximation tool, and that moreover, sometimes a better approximation can be obtained if we use real values outside the interval [0,1]; it is therefore necessary to describe possible extension of t-norms and t-conorms to such new values. It is reasonable to require that this extension be associative, i.e., that the set of truth value with the corresponding operation form a semigroup. Semigroups have been extensively studied in mathematics. In this short paper, we describe several results from semigroup theory which …
Why Fundamental Physical Equations Are Of Second Order?, Takeshi Yamakawa, Vladik Kreinovich
Why Fundamental Physical Equations Are Of Second Order?, Takeshi Yamakawa, Vladik Kreinovich
Departmental Technical Reports (CS)
In this paper, we use a deep mathematical result (namely, a minor modification of Kolmogorov's solution to Hilbert's 13th problem) to explain why fundamental physical equations are of second order. This same result explain why all these fundamental equations naturally lead to non-smooth solutions like singularity.