Computer Engineering Commons^™

Locating The Whole Pattern Is Better Than Locating Its Pieces: A Geometric Explanation Of An Empirical Phenomenon, Scott A. Starks, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical problems, we must find a pattern in an image. For situations in which the desired pattern consists of several simple components, the traditional approach is first to look for such components, and then to see whether the relative locations of these components are consistent with the pattern. Recent experiments have shown that a much more efficient pattern recognition can be achieved if we look for the whole pattern (without decomposing it first). In this paper, we give a simple geometric explanation of this empirical fact.

System Reliability: A Case When Fuzzy Logic Enhances Probability Theory's Ability To Deal With Real-World Problems, Timothy J. Ross, Carlos M. Ferregut, Roberto Osegueda, Vladik Kreinovich

Departmental Technical Reports (CS)

In his recent paper "Probability theory needs an infusion of fuzzy logic to enhance its ability to deal with real-world problems", L. Zadeh explains that probability theory needs an infusion of fuzzy logic to enhance its ability to deal with real-world problems. In this talk, we give an example of a real-world problem for which such an infusion is indeed successful: the problem of system reliability.

From Fuzzy Models To Fuzzy Control, Chitta Baral, Vladik Kreinovich, Hung T. Nguyen, Yeung Yam

Departmental Technical Reports (CS)

Traditional (non-fuzzy) control methodology deals with situations when we know exactly how the system behaves and how it will react to different controls, and we want to choose an appropriate control strategy. This methodology enables us to transform the description of the plant's (system's) behavior into an appropriate control strategy. In many practical situations, we do not have the exact knowledge of the system's behavior, but we have expert-supplied fuzzy rules which describe this behavior. In such situations, it is desirable to transform these description rules into rules describing control. There exist several reasonable heuristics for such transformation; however, the …

Towards Intelligent Virtual Environment For Training Medical Doctors In Surgical Pain Relief, Richard Alo, Kenneth Alo, Vladik Kreinovich

Departmental Technical Reports (CS)

Chronic pain is a serious health problem affecting millions of people worldwide. Spinal cord stimulation is one of the most effective methods of easing the chronic pain. For most patients, a careful selection of weak electric currents drastically decreases the pain level. Engineering progress leads to more and more flexible devices that offer a wide variety of millions of possible simulation regimes. It is not possible to test all of them on each patient, we need an intelligent method of choosing an appropriate simulation regime. In this paper, we describe the need for an intelligent virtual environment for training medical …

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

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 …

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 …

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.

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

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

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

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

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

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

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.

Interval Computations, Soft Computing, And Aerospace Applications, Vladik Kreinovich

Departmental Technical Reports (CS)

No abstract provided.

Justification Of Heuristic Methods In Data Processing Using Fuzzy Theory, With Applications To Detection Of Business Cycles From Fuzzy Data, Vladik Kreinovich, Hung T. Nguyen, Berlin Wu

Departmental Technical Reports (CS)

No abstract provided.

Interval Image Classification Is Np-Hard, Alejandro E. Brito, Vladik Kreinovich

Departmental Technical Reports (CS)

Feature extraction from images to perform object classification is a very hard problem for general solution. We prove that under interval uncertainty, linear classification is NP-hard.

Towards Optimal Pain Relief: Acupuncture And Spinal Cord Stimulation, Richard Alo, Kenneth Alo, Obinna Ilochonwu, Vladik Kreinovich, Hoang Phuong Nguyen

Departmental Technical Reports (CS)

One of the important potential areas of application of intelligent virtual environment is to training medical doctors. One of the main problems in designing the corresponding intelligent system is the computational complexity of the corresponding computational problems. This computational complexity is especially high when the corresponding optimization is a discrete optimization problem, e.g., for pain relief methodologies such as acupuncture and spinal cord stimulation. In this paper, we show how to efficiently solve the corresponding discrete optimization problems. As a result, we get, e.g., a theoretical justification for the heuristic method of "guarded cathode".

Towards Intelligent Virtual Environment For Teaching Telemanipulation Operators: Virtual Tool Approach And Its Interval-Based Justification, L. Olac Fuentes, Vladik Kreinovich

Departmental Technical Reports (CS)

No abstract provided.

Towards Foundations For Traditional Oriental Medicine, Hoang Phuong Nguyen, Scott A. Starks, Vladik Kreinovich

Departmental Technical Reports (CS)

No abstract provided.

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, Vladik Kreinovich, Hung T. Nguyen, Berlin Wu, Krassimir T. Atanassov

Departmental Technical Reports (CS)

No abstract provided.

Np-Hardness In Geometric Construction Problems With One Interval Parameter, Nuria Mata, Vladik Kreinovich

Departmental Technical Reports (CS)

No abstract provided.

Possible New Directions In Mathematical Foundations Of Fuzzy Technology: A Contribution To The Mathematics Of Fuzzy Theory, Hung T. Nguyen, Vladik Kreinovich

Departmental Technical Reports (CS)

No abstract provided.

Complex Problems: Granularity Is Necessary, Granularity Helps, Oscar N. Garcia, Vladik Kreinovich, Luc Longpre, Hung T. Nguyen

Departmental Technical Reports (CS)

No abstract provided.

Uncertainty Representation Explains And Helps Methodology Of Physics And Science In General, Misha Kosheleva, Vladik Kreinovich, Hung T. Nguyen, Bernadette Bouchon-Meunier

Departmental Technical Reports (CS)

No abstract provided.

Cooperative Learning Is Better: Explanation Using Dynamical Systems, Fuzzy Logic, And Geometric Symmetries, Vladik Kreinovich, Edye Johnson-Holubec, Leonid K. Reznik, Misha Kosheleva

Departmental Technical Reports (CS)

No abstract provided.

How To Describe Partially Ordered Preferences: Mathematical Foundations, Olga Kosheleva, Vladik Kreinovich, Hung T. Nguyen, Bernadette Bouchon-Meunier

Departmental Technical Reports (CS)

No abstract provided.

Towards Combining Fuzzy And Logic Programming Techniques, Hung T. Nguyen, Vladik Kreinovich, Daniel E. Cooke, Luqi, Olga Kosheleva

Departmental Technical Reports (CS)

No abstract provided.

From Semi-Heuristic Fuzzy Techniques To Optimal Fuzzy Methods: Mathematical Foundations And Applications, Vladik Kreinovich

Departmental Technical Reports (CS)

Fuzzy techniques have been successfully used in various application areas ranging from control to image processing to decision making. In all these applications, there is usually:

a general idea, and then

there are several possible implementations of this idea; e.g., we can use:

different membership functions,

different "and" and "or" operations,

different defuzzifications, etc.

In the first approximation, the results are usually reasonably robust and independent on this choice, so any heuristic or semi-heuristic choice works OK. However:

if we want to further improve the semi-heuristic "good enough" control or image processing techniques,

we must actually make the selection that …