Engineering Commons^™

Kolmogorov Complexity And Chaotic Phenomena, Vladik Kreinovich, Isaak A. Kunin

Departmental Technical Reports (CS)

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, ...).

Specifically, in addition to the equations, physicists use the following additional difficult-to-formalize property: that the initial conditions and the value of the parameters must not be abnormal. We will describe a natural formalization of this property, and show that this formalization in good accordance with theoretical physics. At present, this formalization has been mainly applied to the foundations of physics. However, potentially, more practical …

Interval Mathematics For Analysis Of Multiresolutional Systems, Vladik Kreinovich, Richard Alo

Departmental Technical Reports (CS)

The more complex the problem, the more complex the system necessary for solving this problem. For very complex problems, it is no longer possible to design the corresponding system on a single resolution level, it becomes necessary to have multiresolutional systems. When analyzing such systems -- e.g., when estimating their performance and/or their intelligence -- it is reasonable to use the multiresolutional character of these systems: first, we analyze the system on the low-resolution level, and then we sharpen the results of the low-resolution analysis by considering higher-resolution representations of the analyzed system. The analysis of the low-resolution level provides …

Fuzzy Measures And Integrals As Aggregation Operators: Solving The Commensurability Problem, Francois Modave, Vladik Kreinovich

Departmental Technical Reports (CS)

The aim of this paper is to shed some light on the use of fuzzy measures and integrals as aggregation operators in multicriteria decision making. These techniques have been widely used on an ad hoc basis, but with no axiomatization. It is possible to obtain preference representation theorems in multicriteria decision making problems, relying on a formal parallelism between decision under uncertainty and multicriteria decision making. Though, it raises some commensurability problems. In this paper, we show how to obtain an axiomatization of multicriteria decision making problems, in a very natural way, and we show how to solve the commensurability …

Selecting A Fuzzy Logic Operation From The Dnf-Cnf Interval: How Practical Are The Resulting Operations?, I. B. Turksen, A. Esper, K. Patel, Scott A. Starks, Vladik Kreinovich

Departmental Technical Reports (CS)

In classical (two-valued) logic, CNF and DNF forms of each propositional formula are equivalent to each other. In fuzzy logic, CNF and DNF forms are not equivalent, they form an interval that contains the fuzzy values of all classically equivalent propositional formulas. If we want to select a single value from this interval, then it is natural to select a linear combination of the interval's endpoints. In particular, we can do that for CNF and DNF forms of "and" and "or", thus designing natural fuzzy analogues of classical "and" and "or" operations. The problem with thus selected "and" and "or" …

Why Is Selecting The Simplest Hypothesis (Consistent With Data) A Good Idea? A Simple Explanation, Vladik Kreinovich, Luc Longpre, Scott Ferson, Lev Ginzburg

Departmental Technical Reports (CS)

No abstract provided.

Main Ideas Behind Owa Lead To A Universal And Optimal Approximation Scheme, Ronald R. Yager, Vladik Kreinovich

Departmental Technical Reports (CS)

Ordered Weighted Averaging (OWA) operators have been successfully applied in many practical problems. We explain this empirical success by showing that these operators are indeed guaranteed to work (i.e., are universal), and that these operators are the best to use (in some reasonable sense).

From Computation With Guaranteed Intervals To Computation With Confidence Intervals: A New Application Of Fuzzy Techniques, Vladik Kreinovich, Hung T. Nguyen, Scott Ferson, Lev Ginzburg

Departmental Technical Reports (CS)

Traditional interval computations provide an estimate for the result y=f(x1,...,xn) of data processing when we know intervals X1,...,Xn that are guaranteed to contain the (unknown) actual values of the quantities x1,...,xn. Often, in addition to these guaranteed intervals, we have confidence intervals for these quantities, i.e., intervals Xi that contain the corresponding values xi with a certain probability. It is desirable, based on the confidence intervals for xi, to produce the resulting confidence interval for y. It turns out that the formulas for computing such resulting confidence interval are closely related with the formulas for processing fuzzy numbers by using …

On Efficient Representation Of Expert Knowledge By Fuzzy Logic: Towards An Optimal Combination Of Granularity And Higher-Order Approaches, Hung T. Nguyen, Vladik Kreinovich

Departmental Technical Reports (CS)

A natural approach to designing an intelligent system is to incorporate expert knowledge into this system. One of the main approaches to translating this knowledge into computer-understandable terms is the approach of fuzzy logic. It has led to many successful applications, but in several aspects, the resulting computer representation is somewhat different from the original expert meaning. Two related approaches have been used to make fuzzy logic more adequate in representing expert reasoning: granularity and higher-order approaches. Each approach is successful in some applications where the other approach did not succeed so well; it is therefore desirable to combine these …

Range Estimation Is Np-Hard For Epsilon-Square Accuracy And Feasible For Epsilon To The Power Of 2-Delta, Vladik Kreinovich

Departmental Technical Reports (CS)

The basic problem of interval computations is: given a function f(x1,...,xn) and n intervals [xi-,xi+], find the (interval) range Y of the given function on the given intervals. It is known that even for quadratic polynomials f(x1,...,xn), this problem is NP-hard. In this paper, following the advice of A. Neumaier, we analyze the complexity of asymptotic range estimation, when the bound "epsilon" on the width of the input intervals tends to 0. We show that for small c>0, if we want to compute the range with an accuracy c times epsilon squared, then the problem is still NP-hard; on …

Computing Variance For Interval Data Is Np-Hard, Scott Ferson, Lev Ginzburg, Vladik Kreinovich, Luc Longpre, Monica Aviles

Departmental Technical Reports (CS)

When we have only interval ranges [xi-,xi+] of sample values x1,...,xn, what is the interval [V-,V+] of possible values for the variance V of these values? We prove that the problem of computing the upper bound V+ is NP-hard. We provide a feasible (quadratic time) algorithm for computing the lower bound V- on the variance of interval data. We also provide a feasible algorithm that computes V+ under reasonable easily verifiable conditions.

Exact Bounds On Sample Variance Of Interval Data, Scott Ferson, Lev Ginzburg, Vladik Kreinovich, Monica Aviles

Departmental Technical Reports (CS)

We provide a feasible (quadratic time) algorithm for computing the lower bound on the sample variance V of interval data. The problem of computing the upper bound on V is, in general, NP-hard. We provide a feasible algorithm that computes the upper bound on V for many reasonable situations.

Detecting And Locating Curved Cracks In Thin Plates By Lamb Wave Reflection: Validated Geometric Approach, Roberto A. Osegueda, Vladik Kreinovich

Departmental Technical Reports (CS)

Lamb waves propagate through a thin plate, and thus, provide a way of scanning this plate and detecting cracks and other faults. The equations describing these waves are rather complex, and, as a result, it is difficult to extract, from the received signal, the location of the fault. Recently, a new geometric approach has been proposed which allows, for linear cracks, to determine the presence and the location of a crack by using only the geometry of wave propagation. In this paper, we extend this approach to a more realistic case of curved cracks, and apply interval techniques to provide …

Optimization Techniques Under Uncertain Criteria, And Their Possible Use In Computerized Education, Vladik Kreinovich, Richard Alo

Departmental Technical Reports (CS)

The existing successful automated computerized systems more or less simulate the way successful human teachers teach. However, computerized systems provide more individualized options that traditional classroom education, and it is desirable to use this additional freedom to further improve the education success rate. In this papers, we briefly overview the experience of a successful Russian training system, and explain how general techniques of optimization under uncertainty can be used to optimize the content development.

On The Optimal Choice Of Quality Metric In Image Compression, Olga Kosheleva, Vladik Kreinovich, Yeung Yam

Departmental Technical Reports (CS)

No abstract provided.

Open-Ended Configurations Of Radio Telescopes: Towards Optimal Design, Vladik Kreinovich, Scott A. Starks, Olga Kosheleva, Andrei Finkelstein

Departmental Technical Reports (CS)

The quality of radio astronomical images drastically depends on where we place the radio telescopes. During the design of the Very Large Array, it was empirically shown that the power law design, in which n-th antenna is placed at a distance n^b from the center, leads to the best image quality. In this paper, we provide a theoretical justification for this empirical fact.

Open-Ended Configurations Of Radio Telescopes: Geometrical Analysis, Vladik Kreinovich, Scott A. Starks, Dima Iourinski, Olga Kosheleva, Andrei Finkelstein

Departmental Technical Reports (CS)

No abstract provided.

Was There Satan's Face In The World Trade Center Fire? A Geometric Analysis, Vladik Kreinovich, Dima Iourinski

Departmental Technical Reports (CS)

Some photos of the 2001 World Trade Center fire reveal a "face" in the smoke which was interpreted, by some people, as the face of Satan. Most journalists believe, however, that the visible smoke configuration can be explained by natural processes, and that the visible "face" is similar to animal shapes that are sometimes observed in the clouds. In this paper, we present a simple geometric analysis that supports this natural-process explanation.

On Symmetric Solution Sets, Goetz Alefeld, Vladik Kreinovich, Guenter Mayer

Departmental Technical Reports (CS)

Given an n x n interval matrix [A] and an interval vector [b] with n components we present an overview on existing results on the solution set S of linear systems of equations Ax=b with symmetric matrices A from [A] and vectors b from [b]. Similarly we consider the set E of eigenpairs associated with the symmetric matrices A from [A]. We report on characterizations of S by means of inequalities, by means of intersection of sets, and by an approach which is generalizable to more general dependencies of the entries. We also recall two methods for enclosing S by …

Probability Of Implication, Logical Version Of Bayes Theorem, And Fuzzy Logic Operations, Hung T. Nguyen, Masao Mukaidono, Vladik Kreinovich

Departmental Technical Reports (CS)

Logical inference starts with concluding that if B implies A, and B is true, then A is true as well. To describe probabilistic inference rules, we must therefore define the probability of an implication "A if B". There exist two different approaches to defining this probability, and these approaches lead to different probabilistic inference rules: We may interpret the probability of an implication as the conditional probability P(A|B), in which case we get Bayesian inference. We may also interpret this probability as the probability of the material implication "A or not B", in which case we get different inference rules. …

Computational Complexity Of Planning With Discrete Time And Continuous State Variables, Chitta Baral, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditionally, most planning research in AI was concentrated on systems whose state can be characterized by discrete-valued fluents. In many practical applications, however, we want to control systems (like robots) whose state can only be described if we used continuous variables (like coordinates). Planning for such systems corresponds, crudely speaking, to Level 2 of the planing language PDDL2.1. In this paper, we analyze the computational complexity of such planning problems.

Automatic Referencing Of Multi-Spectral Images, Roberto Araiza, Hongjei Xie, Scott A. Starks, Vladik Kreinovich

Departmental Technical Reports (CS)

No abstract provided.

Non-Destructive Testing Of Aerospace Structures: Granularity And Data Mining Approach, Roberto Osegueda, Vladik Kreinovich, Lakshmi Potluri, Richard Alo

Departmental Technical Reports (CS)

For large aerospace structures, it is extremely important to detect faults, and non-destructive testing is the only practical way to do it. Based on measurements of ultrasonic waves, Eddy currents, magnetic resonance, etc., we reconstruct the locations of the faults. The best (most efficient) known statistical methods for fault reconstruction are not perfect. We show that the use of expert knowledge-based granulation improves the quality of fault reconstruction.

Uncertainty In Risk Analysis: Towards A General Second-Order Approach Combining Interval, Probabilistic, And Fuzzy Techniques, Scott Ferson, Lev Ginzburg, Vladik Kreinovich, Hung T. Nguyen, Scott A. Starks

Departmental Technical Reports (CS)

Uncertainty is very important in risk analysis. A natural way to describe this uncertainty is to describe a set of possible values of each unknown quantity (this set is usually an interval), plus any additional information that we may have about the probability of different values within this set. Traditional statistical techniques deal with the situations in which we have a complete information about the probabilities; in real life, however, we often have only partial information about them. We therefore need to describe methods of handling such partial information in risk analysis. Several such techniques have been presented, often on …

Use Of Satellite Image Referencing Algorithms To Characterize Asphaltic Concrete Mixtures, Scott A. Starks, Soheil Nazarian, Vladik Kreinovich, Joseph Adidhela

Departmental Technical Reports (CS)

A natural way to test the structural integrity of a pavement is to send signals with different frequencies through the pavement and compare the results with the signals passing through an ideal pavement. For this comparison, we must determine how, for the corresponding mixture, the elasticity E depends on the frequency f in the range from 0.1 to 10^5 Hz. It is very expensive to perform measurements in high frequency area (above 20 Hz). To avoid these measurements, we can use the fact that for most of these mixtures, when we change a temperature, the new dependence changes simply by …

Extending Direct Manipulation In A Text Editor, David G. Novick, Francisco Romero, Edgar Rene Saenz, Armando Sandoval

Departmental Papers (CS)

This paper describes the implementation of a prototype text editor that incorporates conversation-like features through the direct-manipulation modality. In this way, traditional direct-manipulation interaction techniques such as direct reference via pointing can be extended to include techniques more commonly associated with human conversation, such as negotiation of reference. The paper illustrates the use of the prototype with an extended example, and discusses research issues raised by the implementation.

From Planning To Searching For The Shortest Plan: An Optimal Transition, Raul A. Trejo, Joel Galloway, Charanjiv Sachar, Vladik Kreinovich, Chitta Baral, Le Chi Tuan

Departmental Technical Reports (CS)

Since Kautz and Selman's 1992 ECAI paper on satisfiability based planning, there has been several work on planning through finding models of a logical theory. Most of these works focus on finding a plan of a given length. If we want to find the shortest plan, then usually, we try plans of length 1, 2, ..., until we find the first length for which such a plan exists. When the planning problem is difficult and the shortest plan is of a reasonable length, this linear search can take a long time; to speed up the process, it has been proposed …

Statistical And Dempster-Shafer Techniques In Testing Structural Integrity Of Aerospace Structures, Roberto A. Osegueda, Seetharami R. Seelam, Ana C. Holguin, Vladik Kreinovich, Chin-Wang Tao, Hung T. Nguyen

Departmental Technical Reports (CS)

We describe the existing statistics-related methods of testing structural integrity for aerospace structures, describe their drawbacks, how they can be overcome, and compare the resulting techniques.

Representation, Elicitation, And Aggregation Of Uncertainty In Risk Analysis - From Traditional Probabilistic Techniques To More General, More Realistic Approaches: A Survey, Scott Ferson, Vladik Kreinovich

Departmental Technical Reports (CS)

No abstract provided.

A Realistic (Non-Associative) Logic And A Possible Explanations Of 7+-2 Law, Raul A. Trejo, Vladik Kreinovich, I. R. Goodman, Jesus Martinez, Reginaldo Gonzalez

Departmental Technical Reports (CS)

When we know the subjective probabilities (degrees of belief) p1 and p2 of two statements S1 and S2, and we have no information about the relationship between these statements, then the probability of S1&S2 can take any value from the interval [max(p1+p2-1,0),min(p1,p2)]. If we must select a single number from this interval, the natural idea is to take its midpoint. The corresponding "and" operation p1&p_2=(1/2)(max(p1+p2-1,0)+min(p1,p2)) is not associative. However, since the largest possible non-associativity degree |(a&b)&c-a&(b&c)| is equal to 1/9, this non-associativity is negligible if the realistic "granular" degree of belief have granules of width <=1/9. This may explain why humans are most comfortable with <=9 items to choose from (the famous "7 plus minus 2" law).

We also show that the …

How To Interpret Neural Networks In Terms Of Fuzzy Logic?, Sompong Dhompongsa, Vladik Kreinovich, Hung T. Nguyen

Departmental Technical Reports (CS)

Neural networks are a very efficient learning tool, e.g., for transforming an experience of an expert human controller into the design of an automatic controller. It is desirable to reformulate the neural network expression for the input-output function in terms most understandable to an expert controller, i.e., by using words from natural language. There are several methodologies for transforming such natural-language knowledge into a precise form; since these methodologies have to take into consideration the uncertainty (fuzziness) of natural language, they are usually called fuzzy logics.