Computer Engineering Commons^™

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 …

Towards Formalization Of Feasibility, Randomness, And Commonsense Implication: Kolmogorov Complexity, And The Necessity Of Considering (Fuzzy) Degrees, Vladik Kreinovich, Luc Longpre, Hung T. Nguyen

Departmental Technical Reports (CS)

No abstract provided.

Kolmogorov Complexity Justifies Software Engineering Heuristics, Ann Q. Gates, Vladik Kreinovich, Luc Longpre

Departmental Technical Reports (CS)

The "clean bill of health" produced by such a technique does not guarantee that the program is actually correct. In this paper, we show that several heuristic techniques for software testing that have been developed in software engineering can be rigorously justified. In this justification, we use Kolmogorov complexity to formalize the terms "simple" and "random" that these techniques use. The successful formalization of simple heuristics is a good indication that Kolmogorov complexity may be useful in formalizing more complicated heuristics as well.

Beyond Interval Systems: What Is Feasible And What Is Algorithmically Solvable?, Vladik Kreinovich

Departmental Technical Reports (CS)

In many real-life applications of interval computations, the desired quantities appear (in a good approximation to reality) as a solution to a system of interval linear equations. It is known that such systems are difficult to solve (NP-hard) but still algorithmically solvable. If instead of the (approximate) interval linear systems, we consider more realistic (and more general) formulations, will the corresponding problems still be algorithmically solvable? We consider three natural generalizations of interval linear systems: to conditions which are more general than linear systems, to multi-intervals instead of intervals, and to dynamics (differential and difference equations) instead of statics (linear …

Fair Division Under Interval Uncertainty, Ronald R. Yager, Vladik Kreinovich

Departmental Technical Reports (CS)

It is often necessary to divide a certain amount of money between n participants, i.e., to assign, to each participant, a certain portion w(i)>=0 of the whole sum (so that w(1)+...+w(n)=1). In some situations, from the fairness requirements, we can uniquely determine these "weights" w(i). However, in some other situations, general considerations do not allow us to uniquely determine these weights, we only know the intervals [w-(i),w+(i)] of possible fair weights. We show that natural fairness requirements enable us to choose unique weights from these intervals; as a result, we present an algorithm for fair division under interval uncertainty.

Kolmogorov Complexity, Statistical Regularization Of Inverse Problems, And Birkhoff's Formalization Of Beauty, Vladik Kreinovich, Luc Longpre, Misha Kosheleva

Departmental Technical Reports (CS)

Most practical applications of statistical methods are based on the implicit assumption that if an event has a very small probability, then it cannot occur. For example, the probability that a kettle placed on a cold stove would start boiling by itself is not 0, it is positive, but it is so small, that physicists conclude that such an event is simply impossible.

This assumption is difficult to formalize in traditional probability theory, because this theory only describes measures on sets (e.g., for an inverse problem, on the set of all functions) and does not allow us to divide functions …

Multi-Spectral Inverse Problems In Satellite Image Processing, Scott A. Starks, Vladik Kreinovich

Departmental Technical Reports (CS)

Satellite imaging is nowadays one of the main sources of geophysical and environmental information. It is, therefore, extremely important to be able to solve the corresponding inverse problem: reconstruct the actual geophysics- or environment-related image from the observed noisy data.

Traditional image reconstruction techniques have been developed for the case when we have a single observed image. This case corresponds to a single satellite photo. Existing satellites (e.g., Landsat) take photos in several (up to 7) wavelengths. To process this multiple-spectral information, we can use known reasonable multi-image modifications of the existing single-image reconstructing techniques. These modifications, basically, handle each …

Case Study Of Non-Linear Inverse Problems: Mammography And Non-Destructive Evaluation, Olga Kosheleva, S. Cabrera, Roberto A. Osegueda, Carlos M. Ferregut, Soheil Nazarian, M. J. George, Vladik Kreinovich, K. Worden

Departmental Technical Reports (CS)

The inverse problem is usually difficult because the signal (image) that we want to reconstruct is weak. Since it is weak, we can usually neglect quadratic and higher order terms, and consider the problem to be linear. Since the problem is linear, methods of solving this problem are also, mainly, linear (with the notable exception of the necessity to take into consideration, e.g., that the actual image is non-negative).

In most real-life problems, this linear description works pretty well. However, at some point, when we start looking for a better accuracy, we must take into consideration non-linear terms. This may …

A Variation On The Zero-One Law, Andreas Blass, Yuri Gurevich, Vladik Kreinovich, Luc Longpre

Departmental Technical Reports (CS)

Given a decision problem P and a probability distribution over binary strings, for each n, draw independently an instance xn of P of length n. What is the probability that there is a polynomial time algorithm that solves all instances xn of P? The answer is: zero or one.

Adding Fuzzy Integral To Fuzzy Control, Hung T. Nguyen, Vladik Kreinovich, Richard Alo

Departmental Technical Reports (CS)

Sugeno integral was invented a few decades ago as a natural fuzzy-number analogue of the classical integral. Sugeno integral has many interesting applications. It is reasonable to expect that it can be used in all application areas where classical integrals are used, and in many such areas it is indeed useful. Surprisingly, however, it has never been used in fuzzy control, although in traditional control, classical integral is one of the main tools.

In this paper, we show that the appropriately modified Sugeno integral is indeed useful for fuzzy control: namely, it provides numerical characterization of stability and smoothness …

Computational Complexity And Feasibility Of Fuzzy Data Processing: Why Fuzzy Numbers, Which Fuzzy Numbers, Which Operations With Fuzzy Numbers, Hung T. Nguyen, Misha Kosheleva, Olga Kosheleva, Vladik Kreinovich, Radko Mesiar

Departmental Technical Reports (CS)

In many real-life situations, we cannot directly measure or estimate the desired quantity r. In these situations, we measure or estimate other quantities r1,...,rn related to r, and then reconstruct r from the estimates for r_i. This reconstruction is called data processing.

Often, we only have fuzzy information about ri. In such cases, we have fuzzy data processing. Fuzzy data means that instead of a single number ri, we have several numbers that describes the fuzzy knowledge about the corresponding quantity. Since we need to process more numbers, the computation time for fuzzy …

Operations With Fuzzy Numbers Explain Heuristic Methods In Image Processing, Olga Kosheleva, Vladik Kreinovich, Bernadette Bouchon-Meuiner, Radko Mesiar

Departmental Technical Reports (CS)

Maximum entropy method and its heuristic generalizations are very useful in image processing. In this paper, we show that the use of fuzzy numbers enables us to naturally explain these heuristic methods.

Decision Making Based On Satellite Images: Optimal Fuzzy Clustering Approach, Vladik Kreinovich, Hung T. Nguyen, Scott A. Starks, Yeung Yam

Departmental Technical Reports (CS)

In many real-life decision-making situations, in particular, in processing satellite images, we have an enormous amount of information to process. To speed up the information processing, it is reasonable to first classify the situations into a few meaningful classes (clusters), find the best decision for each class, and then, for each new situation, to apply the decision which is the best for the corresponding class. One of the most efficiently clustering methodologies is fuzzy clustering, which is based on the use of fuzzy logic. Usually, heuristic clusterings are used, i.e., methods which are selected based on their empirical efficiency …

Towards The Use Of Aesthetics In Decision Making: Kolmogorov Complexity Formalizes Birkhoff's Idea, Misha Kosheleva, Vladik Kreinovich, Yeung Yam

Departmental Technical Reports (CS)

Decision making is traditionally based on utilitarian criteria such as cost, efficiency, time, etc. These criteria are reasonably easy to formalize; hence, for such criteria, we can select the best decision by solving the corresponding well-defined optimization problem. In many engineering projects, however, e.g., in designing cars, building, airplanes, etc., an important additional criterion which needs to be satisfied is that the designed object should be good looking. This additional criterion is difficult to formalize and, because of that, it is rarely taken into consideration in formal decision making. In the 1930s, the famous mathematician G. D. Birkhoff has proposed …

How To Divide A Territory? A New Simple Differential Formalism For Optimization Of Set Functions, Hung T. Nguyen, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical problems, we must optimize a set function, i.e., find a set A for which f(A) is maximum, where f is a function defined on the class of sets. Such problems appear in design, in image processing, in game theory, etc.

Most optimization problems can be solved (or at least simplified) by using the fact that small deviations from an optimal solution can only decrease the value of the objective function; as a result, some derivative must be equal to 0. This approach has been successfully used, e.g., for set functions in which the desired set A …

Encryption Algorithms Made (Somewhat) More Natural (A Pedagogical Remark), Misha Kosheleva, Vladik Kreinovich, Luc Longpre

Departmental Technical Reports (CS)

Modern cryptographic algorithms, such as DES, IDEA, etc., are very complex and therefore difficult to learn. Textbooks explain in detail how these algorithms work, but they usually do not explain why these algorithms were designed as they were. In this paper, we explain why, and thus, hopefully, make cryptographic algorithms easier to learn.

An Operationalistic Reformulation Of Einstein's Equivalence Principle, Vladik Kreinovich, R. R. Zapatrine

Departmental Technical Reports (CS)

No abstract provided.

A Modification Of Sugeno Integral Describes Stability And Smoothness Of Fuzzy Control, Hung T. Nguyen, Vladik Kreinovich

Departmental Technical Reports (CS)

Sugeno integral was invented a few decades ago as a natural fuzzy analogue of the classical integral. Sugeno integral has many interesting applications. It is reasonable to expect that it can be used in all application areas where classical integrals are used, and in many such areas it is indeed useful. Surprisingly, however, it has never been used in fuzzy control, although in traditional control, classical integral is one of the main tools.

In this paper, we show that the appropriately modified Sugeno integral is indeed useful for fuzzy control: namely, it provides numerical characterization of stability and smoothness of …

Alps: A Logic For Program Synthesis (Motivated By Fuzzy Logic), Daniel E. Cooke, Vladik Kreinovich, Scott A. Starks

Departmental Technical Reports (CS)

One of the typical problems in engineering and scientific applications is as follows: we know the values x1,...,xn of some quantities, we are interested in the values of some other quantities y1,...,ym, and we know the relationships between xi, yj, and, maybe, some auxiliary physical quantities z1,...,zk. For example, we may know an algorithm to compute y2 from x_1, x3, and y1; we may also know an equation F(x1,x2,y1)=0 that relates these values, etc. The question is: can we compute the values of yj, and, if we can, how to do it?

At first glance, this is a problem of …

Where To Bisect A Box? A Theoretical Explanation Of The Experimental Results, Vladik Kreinovich, R. Baker Kearfott

Departmental Technical Reports (CS)

No abstract provided.

Oo Or Not Oo: When Object-Oriented Is Better. Qualitative Analysis And Application To Satellite Image Processing, Ann Q. Gates, Leticia Sifuentes, Scott A. Starks

Departmental Technical Reports (CS)

No abstract provided.

Complexity Of Collective Decision Making Explained By Neural Network Universal Approximation Theorem, Raul A. Trejo, Vladik Kreinovich

Departmental Technical Reports (CS)

No abstract provided.

Strong Negation: Its Relation To Intervals And Its Use In Expert Systems, Scott A. Starks, Vladik Kreinovich, Hung T. Nguyen, Hoang Phuong Nguyen, Mirko Navara

Departmental Technical Reports (CS)

No abstract provided.