Computer Engineering Commons^™

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 …

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 …

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 …

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 …

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.

On How To Merge Sorted Lists Coming From Different Web Search Tools, Ronald R. Yager, Vladik Kreinovich

Departmental Technical Reports (CS)

Different web search tools often complement each other. So, if we want to have a good coverage of all relevant web items, a reasonable strategy is to use different search tools and then merge the resulting lists. How to merge them? In this paper, we describe reasonable axioms for the merging procedure and describe all mergings that satisfy these reasonable axioms.

Optimal Choices Of Potential Functions In Fuzzy Clustering, Vladik Kreinovich, Hung T. Nguyen, Yeung Yam

Departmental Technical Reports (CS)

Fuzzy logic-based clustering techniques are widely used in situations where statistical assumptions are not valid. Whether in estimating cluster centers for model identification purposes or in determining clusters the existing techniques are essentially based upon the choice of some potential functions. As in any design problems of this kind, the choice of such a function has to be justified on a theoretical basis. In this work, we set up a decision frame work and show that optimal potential functions are the ones which are used in current techniques.

Human Visual Perception And Kolmogorov Complexity: Revisited, Vladik Kreinovich, Luc Longpre

Departmental Technical Reports (CS)

Experiments have shown that we can only memorize images up to a certain complexity level, after which, instead of memorizing the image itself, we, sort of, memorize a probability distribution in terms of which this image is "random" (in the intuitive sense of this word), and next time, we reproduce a "random" sample from this distribution. This random sample may be different from the original image, but since it belongs to the same distribution, it, hopefully, correctly reproduces the statistical characteristics of the original image.

The reason why a complex image cannot be accurately memorized is, probably, that our memory …

Run-Time Correctness Checking Is Algorithmically Undecidable For Pointer Data Structures, Mikhail Auguston, Vladik Kreinovich, Luc Longpre

Departmental Technical Reports (CS)

Programs routinely use complicated pointer (linked list-type) data structures such as linked lists, doubly linked lists, different types of trees, etc. These data structures are usually defined {\it inductively}: e.g., a tree can be defined as a structure that results from an empty tree by an arbitrary sequence of adding and deleting elements.

When the program runs, these data structures take dynamically changing shapes. To test program correctness, it is important to check, at run-time, whether a current shape is a correct implementation of the corresponding structure. Algorithms are known for checking the ``shape correctness'' for basic pointer-based data structures …

Telemanipulation: The Virtual Tool Approach And Its Interval-Based Justification, Vladik Kreinovich, Luic Olac Fuentes

Departmental Technical Reports (CS)

No abstract provided.

Application Of Kolmogorov Complexity To Image Compression: It Is Possible To Have A Better Compression, But It Is Not Possible To Have The Best One, Sanjeev Subbaramu, Ann Q. Gates, Vladik Kreinovich

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.

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.

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

Departmental Technical Reports (CS)

No abstract provided.

Interval Approach To Non-Destructive Testing Of Aerospace Structures And To Mammography, Keith Worden, Roberto A. Osegueda, Carlos M. Ferregut, Soheil Nazarian, Eulalio Rodriguez, Debra L. George, Mary J. George, Vladik Kreinovich, Olga Kosheleva, Sergio Cabrera

Departmental Technical Reports (CS)

No abstract provided.

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 …

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 …

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

Departmental Technical Reports (CS)

No abstract provided.

Ordinal Explanation Of The Periodic System Of Chemical Elements, Eric R. Scerri, Vladik Kreinovich, Piotr Wojciechowski, Ronald R. Yager

Departmental Technical Reports (CS)

Textbooks often claim that quantum mechanics explained the periodic system: namely, the actual configuration of electronic orbits that is responsible for the element's chemical properties can be described as the one that minimizes the total energy, and the energy of each configuration can be computed by using quantum mechanics.

However, a careful analysis of this explanation reveals that, in addition to the basic equations of quantum mechanics, we need some heuristic rules that do not directly follow from quantum physics. One reason why additional heuristics are necessary is that the corresponding numerical equations are extremely difficult to solve, and as …

On Geometry Of Radio Antenna Placements, Olga Kosheleva, Vladik Kreinovich, Andrei M. Finkelstein, Steve Chan

Departmental Technical Reports (CS)

No abstract provided.

Coincidences Are Not Accidental: A Theorem, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we formalize and prove the statement that coincidences cannot be accidental, a statement that underlies many useful heuristics in mathematics and physics.

Our proof uses a version of Kolmogorov complexity, a technique originally developed to describe randomness and "accidentalness".

Why Kolmogorov Complexity In Physical Equations, Vladik Kreinovich, Luc Longpre

Departmental Technical Reports (CS)

Several researchers, including M. Gell-Mann, argue that the notion of Kolmogorov complexity, developed in the algorithmic information theory, is useful in physics (i.e., in the description of the physical world). Their arguments are rather convincing, but there seems to be a gap between traditional physical equations and Kolmogorov complexity: namely, it is not clear how the standard equations of physics can lead to algorithmic notions underlying Kolmogorov complexity. In this paper, this "gap" is bridged: we explain how Kolmogorov complexity naturally appear in physical equation.

From Fuzzification And Intervalization To Anglification: A New 5d Geometric Formalism For Physics And Data Processing, Scott A. Starks, Vladik Kreinovich

Departmental Technical Reports (CS)

No abstract provided.

Spinal Cord Stimulation For Chronic Pain Management: Towards An Expert System, Kenneth M. Alo, Richard Alo, Andre De Korvin, Vladik Kreinovich

Departmental Technical Reports (CS)

Chronic pain is a serious health problem affecting millions of people worldwide. Currently, spinal cord simulation is one of the most effective methods of easing the chronic pain. For most patients, a careful selection of weak electric currents enables to drastically decrease the pain level. The first devices offered only a few possible regimes, and it was possible to choose an appropriate regime simply by exhaustive search. Continuous engineering progress leads to more and more flexible devices that offer a wide variety of millions of possible simulation regimes. With this variety, it is no longer possible to test all of …