Computer Science Department

Abstracts of 2014 Reports

Technical Report UTEP-CS-14-79, December 2014

Need for Data Processing Naturally Leads to Fuzzy Logic (and Neural Networks): Fuzzy Beyond Experts and Beyond Probabilities

Vladik Kreinovich, Hung T. Nguyen, and Songsak Sriboonchitta

To appear in *International Journal of Intelligent Systems*

Fuzzy techniques have been originally designed to describe imprecise ("fuzzy") expert knowledge. Somewhat surprisingly, fuzzy techniques have also been successfully used in situations without expert knowledge, when all we have is data. In this paper, we explain this surprising phenomenon by showing that the need for optimal processing of data (including crisp data) naturally leads to fuzzy and neural data processing techniques.

This result shows the potential of fuzzy data processing. To maximally utilize this potential, we need to provide an operational meaning of the corresponding fuzzy degrees. We show that such a meaning can be extracted from the above justification of fuzzy techniques. It turns out that, in contrast to probabilistic uncertainty, the natural operational meaning of fuzzy degrees is indirect -- similarly to the operational meaning of geometry and physics in General Relativity.

Technical Report UTEP-CS-14-78, December 2014

Updated version UTEP-CS_14-78b, July 2015

How Much For an Interval? a Set? a Twin Set? a p-Box? A Kaucher Interval? Towards an Economics-Motivated Approach to Decision Making Under Uncertainty

Joe Lorkowski and Vladik Kreinovich

Published in *Proceedings of the 16th GAMM-IMACS International Symposium on
Scientific Computing, Computer Arithmetic, and Verified Numerical
Computation SCAN'2014*, Wuerzburg, Germany, September 21-26, 2014,
pp. 66-76.

A natural idea of decision making under uncertainty is to assign a fair price to different alternatives, and then to use these fair prices to select the best alternative. In this paper, we show how to assign a fair price under different types of uncertainty.

Original file UTEP-CS-14-78 in pdf

Updated version UTEP-CS-14-78b in pdf

Technical Report UTEP-CS-14-77, December 2014

Revised version UTEP-CS-14-77a, February 2015

Revised version UTEP-CS-14-77b, April 2015

Every Function Computable by an Arithmetic Single Use Expression is a Ratio of Two Multi-Linear Functions: A Theorem

Joe Lorkowski, Olga Kosheleva, and Vladik Kreinovich

Published in *Journal of Uncertain Systems*, 2016, Vol. 10,
No. 1, pp. 48-52.

One of the main problems of interval computation is computing the range of a given function on a given box. In general, computing the exact range is a computationally difficult (NP-hard) problem, but there are important cases when a feasible algorithm for computing such a function is possible. One of such cases is the case of singe-use expressions (SUE), when each variable occurs only once. Because of this, practitioners often try to come up with a SUE expression for computing a given function. It is therefore important to know when such a SUE expression is possible. In this paper, we consider the case of functions that can be computed by using only arithmetic operations (addition, subtraction, multiplication, and division). We show that when there exists a SUE expression for computing such a function, then this function is equal to a ratio of two multi-linear functions (although there are ratios of multi-linear functions for which no SUE expression is possible). Thus, if for a function, no SUE expression is possible, then we should not waste our efforts on finding a SUE expression for computing this function.

Original file UTEP-CS-14-77 in pdf

Revised version UTEP-CS-14-77a in pdf

Revised version UTEP-CS-14-77b in pdf

Technical Report UTEP-CS-14-76, December 2014

Revised version UTEP-CS-14-76b, June 2015

Final version UTEP-CS-17-76c, August 2015

When Can We Reduce Multi-Variable Range Estimation Problem to Two Fewer-Variable Problems?

Joe Lorkowski, Olga Kosheleva, Luc Longpre, and Vladik Kreinovich

Published in *Reliable Computing*, 2015, Vol. 21, pp. 1-10.

Sometimes, a function f of n variables can be represented as a composition of two functions of fewer variables. In this case, the problem of computing the range of f on given intervals can be reduced to two range-computation problems with fewer variables. In this paper, we describe a feasible algorithm that checks whether such a reduction is possible -- and, if it is possible, produces the desired reduction.

Original file UTEP-CS-14-76 in pdf

Revised version UTEP-CS-14-76b in pdf

Final version UTEP-CS-14-76c in pdf

Technical Report UTEP-CS-14-75, December 2014

Revised version UTEP-CS-14-75a, May 2015

50 Years of Fuzzy: from Discrete to Continuous to -- Where?

Vladik Kreinovich, Hung T. Nguyen, Olga Kosheleva, and Rujira Ouncharoen

Published in *Journal of Intelligent and Fuzzy Systems*, 2015,
Vol. 29, pp. 989-1009.

While many objects and processes in the real world are discrete, from the computational viewpoint, discrete objects and processes are much more difficult to handle than continuous ones. As a result, a continuous approximation is often a useful way to describe discrete objects and processes. We show that the need for such an approximation explains many features of fuzzy techniques, and we speculate on to which promising future directions of fuzzy research this need can lead us.

Original file UTEP-CS-14-75 in pdf

Revised version UTEP-CS-14-75a in pdf

Technical Report UTEP-CS-14-74, December 2014

Interval computations and interval-related statistical techniques: estimating uncertainty of the results of data processing and indirect measurements

Vladik Kreinovich

Published in: Franco Pavese,
Wolfram Bremser, Anna Chunovkina, Nicolas Fisher, and Alistair B.
Forbes (eds.),
*Advanced Mathematical and Computational Tools in Metrology and
Testing AMTCM'X*, World Scientific, Singapore, 2015, pp. 38-49.

In many practical situations, we only know the upper bound Δ on the measurement error: |Δx| ≤ Δ. In other words, we only know that the measurement error is located on the interval [−Δ, Δ]. The traditional approach is to assume that Δx is uniformly distributed on [−Δ, Δ]. In some situations, however, this approach underestimates the error of indirect measurements. It is therefore desirable to directly process this interval uncertainty. Such "interval computations" methods have been developed since the 1950s. In this paper, we provide a brief overview of related algorithms and results.

Technical Report UTEP-CS-14-73, December 2014

Among Several Successful Algorithms, Simpler Ones Usually Work Better: A Possible Explanation of an Empirical Observation

Vladik Kreinovich and Olga Kosheleva

Published in *Mathematical Structures and Modeling*, 2015,
Vol. 33, pp. 50-55.

Often, several different algorithms can solve a certain practical problem. Sometimes, algorithms which are successful in solving one problem can solve other problems as well. How can we decide which of the original algorithms is the most promising -- i.e., which is more probable to be able to solve other problem? In many cases, the simplest algorithms turns out to be the most successful. In this paper, we provide a possible explanation for this empirical observation.

Technical Report UTEP-CS-14-72, December 2014

Newton's Laws: What is Their Operational Meaning?

Olga Kosheleva and Vladik Kreinovich

Published in *Mathematical Structures and Modeling*, 2015,
Vol. 33, pp. 38-49.

Newton's mechanics is one of the most successful theories in the history of science; its success is based on three Newton's laws. At first glance, the Newton's laws that describe the relation between masses, forces, and accelerations are very clear and straightforward. However, the situation becomes more ambiguous if we take into account that the notions of mass and force are not operationally defined. In this paper, we describe the operational meaning of Newton's laws.

Technical Report UTEP-CS-14-71, December 2014

Updated version UTEP-CS-14-71a, January 2015

Minimax Portfolio Optimization under Interval Uncertainty

Meng Yuan, Xu Lin, Junzo Watada, and Vladik Kreinovich

Published in *Journal of Advanced Computational Intelligence and
Intelligent Informatics (JACIII)*, 2015, Vol. 19, No. 5,
pp. 575-580.

In the 1950s, Markowitz proposed to combine different investment instruments to design a portfolio that either maximizes the expected return under constraints on volatility (risk) or minimizes the risk under given expected return. Markowitz's formulas are still widely used in financial practice. However, these formulas assume that we know the exact values of expected return and variance for each instrument, and that we know the exact covariance of every two instruments. In practice, we only know these values with some uncertainty. Often, we only know the bounds of these values -- i.e., in other words, we only know the intervals that contain these values. In this paper, we show how to select an optimal portfolio under such interval uncertainty.

Original file in pdf

updated version in pdf

Technical Report UTEP-CS-14-70, November 2014

Mike Loya Center for Innovation and Commerce Grand Challenge Workshop Report

Technical Report UTEP-CS-14-69, November 2014

Updated version UTEP-CS-14-69b, September 2015

Updated version UTEP-CS-14-69c, July 2024

Asymptotically Tight Algorithm for Checking Whether a Given Vector Is a Solution to a Given Interval-Quantifier Linear System

Vladik Kreinovich

In many practical situations, we have a linear dependence between different quantities. In such situations, we often need to solve the corresponding systems of linear equations. Often, we know the parameters of these equations with interval uncertainty. In this case, depending on the practical problem, we have different notions of a solution. For example, if we determine parameters from observations, we are interested in all the unknowns which satisfy the given system of linear equations for some possible values of the parameters. If we design a system so that it does not exceed given tolerance bounds, then we need to make sure that for all possible values of the design parameters there exist possible values of the outcome parameters for which the system is satisfied, etc. In general, we can have an arbitrary sequence of quantifiers corresponding to different parameters. The resulting systems are known as interval-quantifier linear systems.

In this paper, we provide an asymptotically tight algorithm for checking whether a given vector is a solution to a given interval-quantifier linear system. For a system of m equations with n unknown, this algorithm takes time Θ(m * n).

Original file UTEP-CS-14-69 in pdf

Updated version UTEP-CS-14-69b in pdf
Updated version UTEP-CS-14-69c in pdf

Technical Report UTEP-CS-14-68, October 2014

How to Gauge Unknown Unknowns: A Possible Theoretical Explanation of the Usual Safety Factor of 2

Joe Lorkowski and Vladik Kreinovich

Published in *Mathematical Structures and Modeling*, 2014,
Vol. 32, pp. 49-52.

To gauge the accuracy of a measuring instrument, engineers analyze
possible factors contributing to the instrument's inaccuracy. In
addition to *known* factors, however, there are usually
*unknown* factors which also contribute to the instrument's
inaccuracy. To properly gauge the instrument's accuracy -- and
thus, to make sure that we do not compromise our safety by
underestimating the inaccuracy -- we need to also take these
"unknown unknowns" into account. In practice, this is usually
done by multiplying the original estimate for inaccuracy by a
"safety" factor of 2. In this paper, we provide a possible
theoretical explanation for this empirical factor.

Technical Report UTEP-CS-14-67, October 2014

Granularity Explains Empirical Factor-of-Three Relation Between Probabilities of Pulmonary Embolism in Different Patient Categories

Beverly Rivera, Francisco Zapata, and Vladik Kreinovich

Published in *Mathematical Structures and Modeling*, 2014,
Vol. 32, pp. 130-135.

Pulmonary embolism is a very dangerous difficult-to-detect medical condition. To diagnose pulmonary embolism, medical practitioners combine indirect signs of this condition into a single score, and then classify patients into low-probability, intermediate-probability, and high-probability categories. Empirical analysis shows that, when we move from each category to the next one, the probability of pulmonary embolism increases by a factor of three. In this paper, we provide a theoretical explanation for this empirical relation between probabilities.

Technical Report UTEP-CS-14-66, October 2014

Examining the Consistence of Futures Margin Levels using Bivariate Extreme Value Copulas

X. Gong, H. T. Nguyen, V. Kreinovich, and S. Sriboonchitta

Published in *Thai Journal of Mathematics*, 2014, Special
Issue on Copula Mathematics and Econometrics, pp. 39-57.

This study examines the consistence of the futures margin levels of different commodities and combinations in the CME group by Extreme Value Copula (EVC). We find that if we ignore the co-movements of the commodities, the margins become consistent with each other, and the margin violation rates hover around 0.5%. However, if we consider the co-movement of the related commodities using EVC, the margin levels are found to be not consistent anymore, especially in the combinations of strongly related commodities which are in the same category. Therefore, we suggest that the CME group should try to harmonize the margins policy with respect to the dependence between the futures in the future.

Technical Report UTEP-CS-14-65, September 2014

A Catalog of While Loop Specification Patterns

Aditi Barua and Yoonsik Cheon

This document provides a catalog of while loop patterns along with their skeletal specifications. The specifications are written in a functional form known as intended functions. The catalog can be used to derive specifications of while loops by first matching the loops to the cataloged patterns and then instantiating the skeletal specifications of the matched patterns. Once their specifications are formulated and written, the correctness of while loops can be proved rigorously or formally using the functional program verification technique in which a program is viewed as a mathematical function from one program state to another.

Technical Report UTEP-CS-14-64, September 2014

Updated version UTEP-CS-14-64a, November 2014

Is the World Itself Fuzzy? Physical Arguments for -- and Unexpected Computational Consequences of -- Zadeh's Vision

Vladik Kreinovich and Olga Kosheleva

Published in: Dan E. Tamir, Naphtali David Rishe, and Abraham
Kandel (eds.), *Fifty Years of Fuzzy Logic and Its Applications*,
Springer-Verlag, Berlin, Heidelberg, 2015, pp. 297-313.

Fuzzy methodology has been invented to describe imprecise ("fuzzy") human statements about the world, statements that use imprecise words from natural language like "small" or "large". Usual applications of fuzzy techniques assume that the world itself is ``crisp'', that there are exact equations describing the world, and fuzziness of our statements is caused by the incompleteness of our knowledge. But what if the world itself is fuzzy? What if there is no perfect system of equations describing the physical world -- in the sense that no matter what system of equations we try, there will always be cases when this system will lead to wrong predictions? This is not just a speculation: this idea is actually supported by many physicists. At first glance, this is a pessimistic idea: no matter how much we try, we will never be able to find the Ultimate Theory of Everything. But it turns out that this idea also has its optimistic aspects: namely, in this chapter, we show (somewhat unexpectedly), that if such a no-perfect-theory principle is true, then the use of physical data can drastically enhance computations.

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Technical Report UTEP-CS-14-63, September 2014

Formalizing the Informal, Precisiating the Imprecise: How Fuzzy Logic Can Help Mathematicians and Physicists by Formalizing Their Intuitive Ideas

Olga Kosheleva, Renata Reiser, and Vladik Kreinovich

Published in: Rudolf Seising, Enric Trillas, and Janusz
Kacprycz (eds.), *Towards the Future of Fuzzy Logic*, Springer
Verlag, 2015, pp. 301-321.

Fuzzy methodology transforms expert ideas -- formulated in terms of words from natural language -- into precise rules and formulas. In this paper, we show that by applying this methodology to intuitive physical and mathematical ideas, we can get known fundamental physical equations and known mathematical techniques for solving these equations. This fact makes us confident that in the future, fuzzy techniques will help physicists and mathematicians to transform their imprecise ideas into new physical equations and new techniques for solving these equations.

Technical Report UTEP-CS-14-62, August 2014

Updated version UTEP-CS-14-62a, December 2014

Granularity Helps Explain Seemingly Irrational Features of Human Decision Making

Joe Lorkowski and Vladik Kreinovich

Published in:
Witold Pedrycz and Shyi-Ming Chen (eds.), *Granular Computing
and Decision-Making: Interactive and Iterative Approaches*,
Springer Verlag, Cham, Switzerland, 2015, pp. 1-31.

Starting from well-known studies by Kahmenan and Tarsky,
researchers have found many examples when our decision making --
and our decision making -- seem to be irrational. In this chapter,
we show that this seemingly irrational decision making can be
explained if we take into account that human abilities to process
information are limited; as a result, instead of the exact
*values* of different quantities, we operate with *granules*
that contain these values. On several examples, we show that
optimization under such granularity restriction indeed leads to
observed human decision making. Thus, granularity helps explain
seemingly irrational human decision making.

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Technical Report UTEP-CS-14-61, August 2014

From Numerical Probabilities to Linguistic Probabilities: A Theoretical Justification of Empirical Granules Used in Risk Management

Beverly Rivera, Francisco Zapata, and Vladik Kreinovich

Published in *Applied Mathematical Sciences*, 2014, Vol. 8,
No. 144, pp. 7195-7200.

In many risk management situations, instead of the exact probability values, specialists use a granule to which this probability belongs. Specifically, they use five granules, corresponding to thresholds 10%, 40%, 60%, and 90%. In this paper, we provide an explanation for such non-uniform granulation.

Technical Report UTEP-CS-14-60, August 2014

Revised version UTEP-CS-14-60b, June 2015

Security Risk Assessment: Towards a Justification for the Security Risk Factor Table Model

Beverly Rivera, Francisco Zapata, and Vladik Kreinovich

Published in *Journal of Advanced Computational Intelligence and
Intelligent Informatics (JACIII)*, 2015, Vol. 19, No. 5,
pp. 676-680.

One of the widely used methods to gauge risk is the Security Risk Factor Table (SRFT) model. While this model has been empirically successful, its use is limited by the fact that its formulas do not have a theoretical explanation -- and thus, there is no guarantee that these formulas will work in other situations as well. In this paper, we provide a theoretical explanation for the SFRT formulas.

Original file UTEP-CS-14-60 in
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Revised version UTEP-CS-14-60b in
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Technical Report UTEP-CS-14-59, August 2014

Log-Periodic Power Law as a Predictor of Catastrophic Events: A New Mathematical Justification

Vladik Kreinovich, Hung T. Nguyen, and Songsak Sriboonchitta

Published in *Proceedings of the International Conference on Risk
Analysis in Meteorological Disasters RAMD'2014*,
Nanjing, China, October 12-13, 2014

To decrease the damage caused by meteorological disasters, it is important to be able to predict these disasters as accurately as possible. One of the most promising ways of achieving such a prediction comes from the observation that in the vicinity of a catastrophic event, many parameters exhibit log-periodic power behavior, with oscillations of increasing frequency. By fitting the corresponding formula to the observations, it is often possible to predict the catastrophic event. Such successful predictions were made in many application areas ranging from ruptures of fuel tanks to earthquakes to stock market disruptions. The fact that similar formulas can be applied to vastly different systems seems to indicate that the log-periodic power behavior is not related to a specific nature of the system, it is caused by general properties of system. In this paper, we indeed provide a general system-based explanation of this law. The general character of this explanation makes us confident that this law can be also used to predict meteorological disasters.

Technical Report UTEP-CS-14-58, August 2014

Kekule's Benzene Structure: A Case Study of Teaching Usefulness of Symmetry

Olga Kosheleva and Vladik Kreinovich

Published in *Applied Mathematical Sciences*, 2014, Vol. 8,
No. 144, pp. 7183-7194.

Benzene is one of the basic building blocks of organic molecules. One of the reasons for benzene's ubiquity is its unusual ring structure first discovered by Kekule in 1865. In this paper, we show that a simple symmetry-based analysis can narrow down possible benzene structures to three ring ones, including the Kekule's ring. Thus, Kekule's benzene structure provides a good pedagogical example on which one can explain usefulness of symmetries.

Technical Report UTEP-CS-14-57, August 2014

Updated version UTEP-CS-14-57a, January 2015

Why Lattice-Valued Fuzzy Values? A Mathematical Justification

Rujira Ouncharoen, Vladik Kreinovich, and Hung T. Nguyen

Published in *Journal of Intelligent and Fuzzy Systems*, 2015,
Vol. 29, No. 4, pp. 1421-1425.

To take into account that expert's degrees of certainty are not always comparable, researchers have used partially ordered set of degrees instead of the more traditional linearly (totally) ordered interval [0,1]. In most cases, it is assumed that this partially ordered set is a lattice, i.e., every two elements have the greatest lower bound and the least upper bound. In this paper, we prove a theorem explaining why it is reasonable to require that the set of degrees is a lattice.

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Technical Report UTEP-CS-14-56, August 2014

Towards a Formal Description of Understandability (Causality, Pre-Requisites): From Prosorov's Phonocentric Topology to More General Interior (Closure) Structures

Olga Kosheleva and Vladik Kreinovich

Published in *Mathematical Structures and Modeling*, 2014,
Vol. 31, pp. 18-26.

In many real life situations, a text consists of related parts;
so, to understand a part, we need to first understand some (or
all) preceding parts: e.g., to understand Chapter 3, we first need
to understand Chapters 1 and 2. In many cases, this dependence is
described by a partial order. For this case, O.~Prosorov proposed
a natural description of the dependence structure as a topology
(satisfying the separation axiom T_{0}).

In some practical situations, dependence is more general than
partial order: e.g., to understand Chapter 3, we may need to
understand either Chapter 1 or Chapter 2, but it is not necessary
to understand both. We show that such a general dependence can be
naturally described by a known generalization of topology: the
notion of an interior (or, equivalently, closure) structure
(provided, of course, that this structure satisfies a natural
analog of T_{0}-separability).

Technical Report UTEP-CS-14-55, August 2014

If Many Physicists Are Right and No Physical Theory Is Perfect, Then by Using Physical Observations, We Can Feasibly Solve Almost All Instances of Each NP-Complete Problem

Olga Kosheleva, Michael Zakharevich, and Vladik Kreinovich

Published in *Mathematical Structures and Modeling*, 2014,
Vol. 31, pp. 4-17.

Many real-life problems are, in general, NP-complete, i.e., informally speaking, are difficult to solve -- at least on computers based on the usual physical techniques. A natural question is: can the use of non-standard physics speed up the solution of these problems? This question has been analyzed for several specific physical theories, e.g., for quantum field theory, for cosmological solutions with wormholes and/or casual anomalies, etc. However, many physicists believe that no physical theory is perfect, i.e., that no matter how many observations support a physical theory, inevitably, new observations will come which will require this theory to be updated. In this paper, we show that if such a no-perfect-theory principle is true, then the use of physical data can drastically speed up the solution of NP-complete problems: namely, we can feasibly solve almost all instances of each NP-complete problem.

Technical Report UTEP-CS-14-54, August 2014

Homotopy Techniques in Solving Systems of Nonlinear Equations: A Theoretical Justification of Convex Combinations

Nicholas Sun

Published in *Journal of Uncertain Systems*, 2016, Vol. 10,
No. 1, pp. 68-71.

One of the techniques for solving systems of non-linear equations
F_{1}(x_{1},...,x_{n}) = 0, ...,
F_{n}(x_{1},...,x_{n}) = 0,
(F(x) = 0 in vector notations) is a *homotopy method*, when we
start with
a solution of a simplified (and thus easier-to-solve) approximate
system G_{i}(x_{1},...,x_{n}) = 0,
and then gradually adjust this
solution by solving intermediate systems of equation
H_{i}(x_{1},...,x_{n}) = 0 for an
appropriate "transition" function
H(x) = f(λ,F(x),G(x)).
The success of this method depends on the selection of the
appropriate combination function
f(λ,u_{1},u_{2}).
The most commonly used combination function is the *convex
homotopy* function f(λ,u_{1},u_{2}) =
λ * u_{1} +
(1 − λ) * u_{2}. In this
paper, we provide a theoretical justification for this combination
function.

Technical Report UTEP-CS-14-53, July 2014

Updated version UTEP-CS-14-54b, July 2024

How to Estimate Forecasting Quality: A System-Motivated Derivation of Symmetric Mean Absolute Percentage Error (SMAPE) and Other Similar Characteristics

Vladik Kreinovich, Hung T. Nguyen, and Rujira Ouncharoen

When comparing how well different algorithms forecast time series, researchers use an average value of the ratio |x-y|/(|x|+|y|)/2), known as the Symmetric Mean Absolute Percentage Error (SMAPE). In this paper, we provide a system-motivated explanation for this formula. We also explain how this formula explains the use of geometric mean to combine different forecasts.

Original file UTEP-CS-14-53 in pdf

Updated version UTEP-CS-14-53b in pdf

Technical Report UTEP-CS-14-52, July 2014

Updated version UTEP-CS-14-52a, June 2024

Increased Climate Variability Is More Visible Than Global Warming: A General System-Theory Explanation

L. Octavio Lerma, Craig Tweedie, and Vladik Kreinovich

To appear in *Proceedings of the
9th World Conference on Soft Computing*, Baku, Azerbaijan,
September 24--27, 2024, to appear.

While global warming is a statistically confirmed long-term phenomenon, its most visible consequence is not the warming itself but, somewhat surprisingly, the increased climate variability. In this paper, we use the general system theory ideas to explain why increased climate variability is more visible than the global warming itself.

Original file UTEP-CS-14-52 in
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Technical Report UTEP-CS-14-51, June 2014

Updated version UTEP-CS-14-51a, July 2014

Using Second-Order Probabilities to Make Maximum Entropy Approach to Copulas More Reasonable

Hung T. Nguyen, Vladik Kreinovich, and Berlin Wu

Published in *Thai Journal of Mathematics*, 2014, Special
Issue on Copula Mathematics and Econometrics, pp. 1-10.

Copulas are a general way of describing dependence between two or more random variables. When we only have partial information about the dependence, i.e., when several different copulas are consistent with our knowledge, it is often necessary to select one of these copulas. A frequently used method of selecting this copula is the maximum entropy approach, when we select a copula with the largest entropy. However, in some cases, the maximum entropy approach leads to an unreasonable selection -- e.g., even if we know that the two random variables are positively correlated, the maximum entropy approach completely ignores this information. In this paper, we show how to properly modify the maximum entropy approach so that it will lead to more reasonable results: by applying this approach not to the probabilities themselves, but to "second order" probabilities -- i.e., probabilities of different probability distributions.

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Technical Report UTEP-CS-14-50, June 2014

How to Modify Grade Point Average (GPA) to Make It More Adequate

Joe Lorkowski, Olga Kosheleva, and Vladik Kreinovich

Published in *International Mathematical Forum*, 2014,
Vol. 9, No. 28, pp. 1363-1367.

At present, the amounts of knowledge acquired by different graduates of the same program are usually compared by comparing their GPAs. We argue that this is not always the most adequate description: for example, if, after completing all required classes with the highest grade of "excellent" (A), a student takes an additional challenging class and gets a "satisfactory" grade (C), the amount of her knowledge increases, but the GPA goes down. We propose a modification of the GPA which is free of this drawback and is, thus, more adequate for describing the student's knowledge. We also provide a psychological explanation for why people cling to the traditional GPA.

Technical Report UTEP-CS-14-49, June 2014

Towards Fast and Reliable Localization of an Underwater Object: An Interval Approach

Quentin Brefort, Luc Jaulin, Martine Ceberio, and Vladik Kreinovich

Published in *Journal of Uncertain Systems*, 2015, Vol. 9,
No. 2, pp. 95-102.

To localize an underwater object, we measure the distance to this object from several sonar sensors with known locations. The problem is that the signal sent by some of the sonars is reflected not by the desired object(s), but by some auxiliary object and thus, the values measured by these sensors are drastically different from the distance to the desired object. To solve this problem, currently probabilistic methods are used; however, since we do not know the exact probability distributions, these methods may miss the actual location of the object. There exist interval-based methods which provide guaranteed (reliable) bounds on the object's location, but these methods sometimes require too much computation time. In this paper, we propose a new faster algorithm for reliable localization of underwater objects.

Technical Report UTEP-CS-14-48, June 2014

Updated version UTEP-CS-14-48a, October 2014>br> Likert-type fuzzy uncertainty from a traditional decision making viewpoint: how symmetry helps explain human decision making (including seemingly irrational behavior)

Joe Lorkowski and Vladik Kreinovich

Published in *Applied and Computational Mathematics*, 2014,
Vol. 13, No. 3, pp. 275-298.

One of the main methods for eliciting the values of the membership function μ(x) is to use the Likert-type scales, i.e., to ask the user to mark his or her degree of certainty by an appropriate mark k on a scale from 0 to n and take μ(x) = k/n. In this paper, we show how to describe this process in terms of the traditional decision making, and we conclude that the resulting membership degrees incorporate both probability and utility information. It is therefore not surprising that fuzzy techniques often work better than probabilistic techniques (which only take into account the probability of different outcomes). We also show how symmetry helps explain human decision making, including seemingly irrational behavior.

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Technical Report UTEP-CS-14-47, June 2014

Updated version UTEP-CS-14-47b, October 2014

Why Ricker Wavelets Are Successful in Processing Seismic Data: Towards a Theoretical Explanation

Afshin Gholamy and Vladik Kreinovich

Published in *Proceedings of the IEEE Symposium on Computational
Intelligence for Engineering Solutions CIES'2014*, Orlando, Florida,
December 9-12, 2014, pp. 11-16.

In many engineering applications ranging from engineering seismology to petroleum engineering and civil engineering, it is important to process seismic data. In processing seismic data, it turns out to be very efficient to describe the signal's spectrum as a linear combination of Ricker wavelet spectra. In this paper, we provide a possible theoretical explanation for this empirical efficiency. Specifically, signal propagation through several layers is discussed, and it is shown that the Ricker wavelet is the simplest non-trivial solution for the corresponding data processing problem, under the condition that the described properties of the approximation family are satisfied.

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Technical Report UTEP-CS-14-46, June 2014

How to Explain the Definition of Stochastic Affiliation to Economics Students

Tonghui Wang, Olga Kosheleva, and Vladik Kreinovich

Published in *Journal of Uncertain Systems*, 2015, Vol. 9,
No. 2, pp. 148-150.

To formally describe the intuitive idea of "positive correlation" between two quantities, it is often helpful to use the notion of stochastic affiliation. While this notion is useful, its usual definition is not intuitively clear -- which make it difficult to explain this notion to, e.g., economics students. To help students understand this notion, in this paper, we show how the notion of stochastic affiliation can be explained in clear probabilistic terms.

Technical Report UTEP-CS-14-45, June 2014

Possible Geometric Explanations for Basic Empirical Dependencies of Systems Engineering

Francisco Zapata and Vladik Kreinovich

Published in *Journal of Uncertain Systems*, 2015, Vol. 9,
No. 2, pp. 151-155.

In this paper, we provide possible geometric explanation for basic empirical dependencies of system engineering: that a properly designed system should have no more than 7 plus minus 2 elements reporting to it, and that the relative cost of correcting a defect on different stages of the system's life cycle is 3--6 on the second (design) stage, 20--100 on the third (development) stage, and 250--1000 on the fourth (production and testing) stage.

Technical Report UTEP-CS-14-44, June 2014

Updated version UTEP-CS-14-44a, July 2014

Constructive Mathematics in St. Petersburg, Russia: A (Somewhat Subjective) View from Within

Vladik Kreinovich

Published in: Francine F. Abeles and Mark E. Fuller,
*Modern Logic 1850-1950. East and West,*
Birkhauser, Basel, 2016, pp. 205-236.

In the 1970 and 1980s, logic and constructive mathematics were an important part of my life; it's what I defended in my Master's thesis, it was an important part of my PhD dissertation. I was privileged to work with the giants. I visited them in their homes. They were who I went to for advice. And this is my story.

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Technical Report UTEP-CS-14-43, June 2014

Observable Causality Implies Lorentz Group: Alexandrov-Zeeman-Type Theorem for Space-Time Regions

Olga Kosheleva and Vladik Kreinovich

Published in *Mathematical Structures and Modeling*, 2014,
Vol. 30, pp. 4-14.

The famous Alexandrov-Zeeman theorem proves that causality implies Lorentz group. The physical meaning of this result is that once we observe which event can causally affect which other events, then, using only this information, we can reconstruct the linear structure of the Minkowski space-time. The original Alexandrov-Zeeman theorem is based on the causality relation between events represented by points in space-time. Knowing such a point means that we know the exact moment of time and the exact location of the corresponding event - and that this event actually occurred at a single moment of time and at a single spatial location. In practice, events take some time and occupy some spatial area. Besides, even if we have a point-wise event, we would not be able to know the exact moment of time and exact spatial location - since the only way to determine the moment of time and the spatial location is by measurement, and measurements are never absolutely accurate. To come up with a more realistic description of observable causality relation between events, we need to consider events which are not pointwise, but rather represented by bounded regions A in the Minkowski space-time. When we have two events represented by regions A and B, the fact that we have observed that the first event can causally influence the second one means that a causally precedes b for some points a from A and b from B. In this paper, we show that even if we only know the causal relation between such regions, we can still reconstruct the linear structure on the Minkowski space-time. Thus, already observable causality implies Lorentz group.

Technical Report UTEP-CS-14-42, June 2014

Updated version UTEP-CS-14-42a, October 2014

If We Take Into Account that Constraints Are Soft, Then Processing Constraints Becomes Algorithmically Solvable

Quentin Brefort, Luc Jaulin, Martine Ceberio, and Vladik Kreinovich

Published in *Proceedings of the IEEE Symposium on Computational
Intelligence for Engineering Solutions CIES'2014*, Orlando, Florida,
December 9-12, 2014, pp. 1-10.

Constraints are ubiquitous in science and engineering. Constraints
describe the available information about the state of the system,
constraints describe possible relation between current and future states
of the system, constraints describe which future states we would like to
obtain. To solve problems from engineering and science, it is therefore
necessary to process constraints. We show that if we treat constraints as *hard*
(*crisp*), with all the threshold values exactly known, then in the general case,
all the corresponding computational problems become algorithmically unsolvable.
However, these problems become algorithmically solvable if we take into
account that in reality,
constraints are *soft*: we do not know the exact values of the corresponding
thresholds, we do not know the exact dependence between the present and future states, etc.

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Technical Report UTEP-CS-14-41, May 2014

A new reconstruction from the F-transform components

Michal Holcapek, Irina Perfiieva, and Vladik Kreinovich

Published in *Fuzzy Sets and Systems*, 2016, Vol. 288,
pp. 3-25.

According to a sampling theorem, any band-limited and continuous signal can be uniquely reconstructed from certain of its values. We show that a reconstruction can be obtained from the set of F-transform components and moreover, the sampling theorem follows as a particular case. A special attention is paid to the case where sample values of a signal come with noise. We show that in the presence of noise, a more accurate reconstruction than that based on the sampling theorem can be obtained, if instead of noised sample values the F- transform components of the signal with respect to a generalized fuzzy partition are used.

Technical Report UTEP-CS-14-40, May 2014

Updated version UTEP-CS-14-40a, December 2015

A Simple Probabilistic Explanation of Term Frequency-Inverse Document Frequency (tf-idf) Heuristic (and Variations Motivated by This Explanation)

Lukas Havrlant and Vladik Kreinovich

Published in *International Journal of General Systems*, 2017,
Vol. 46, No. 1, pp. 27-36.

In document analysis, an important task is to automatically find keywords which best describe the subject of the document. One of the most widely used techniques for keyword detection is a technique based on the term frequency-inverse document frequency (tf-idf) heuristic. This techniques has some explanations, but these explanations are somewhat too complex to be fully convincing. In this paper, we provide a simple probabilistic explanation for the tf-idf heuristic. We also show that the ideas behind explanation can help us come up with more complex formulas which will hopefully lead to a more adequate detection of keywords.

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Technical Report UTEP-CS-14-39, May 2014

Updated version UTEP-CS-14-39a, July 2014

Extended version UTEP-CS-14-39b, November 2015

From Mean and Median Income to the Most Adequate Way of Taking Inequality Into Account

Vladik Kreinovich, Hung T. Nguyen, and Rujira Ouncharoen

Original version UTEP-CS-14-39a published in: Van-Nam Huynh,
Vladik Kreinovich, Songsak Sriboonchitta, and
Komsan Suriya (eds.), *Econometrics of Risk*, Springer Verlag, Berlin,
Heidelberg, 2015, pp. 63-73.

How can we compare the incomes of two different countries or
regions? At first glance, it is sufficient to compare the mean
incomes, but this is known to be not a very adequate comparison:
according to this criterion, a very poor country with a few
super-rich people may appear to be in good economic shape. A more
adequate description of economy is the *median* income.
However, the median is also not always fully adequate: e.g.,
raising the income of very poor people clearly improves the overall
economy but does not change the median. In this paper, we use known
techniques from group decision making -- namely, Nash's bargaining
solution -- to come up with the most adequate measure of "average"
income: geometric mean. On several examples, we illustrate how this
measure works.

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Technical Report UTEP-CS-14-38, May 2014

Updated version UTEP-CS-14-38a, July 2014

What If We Only Have Approximate Stochastic Dominance?

Vladik Kreinovich, Hung T. Nguyen, and Songsak Sriboonchitta

Published in: Van-Nam Huynh, Vladik Kreinovich, Songsak Sriboonchitta, and
Komsan Suriya (eds.) *Econometrics of Risk*,
Springer Verlag, Berlin, Heidelberg, 2015, pp. 53-61.

In many practical situations, we need to select one of the two alternatives, and we do not know the exact form of the user's utility function -- e.g., we only know that it is increasing. In this case, stochastic dominance result says that if the cumulative distribution function (cdf) corresponding to the first alternative is always smaller than or equal than the cdf corresponding to the second alternative, then the first alternative is better. This criterion works well in many practical situations, but often, we have situations when for most points, the first cdf is smaller but at some points, the first cdf is larger. In this paper, we show that in such situations of approximate stochastic dominance, we can also conclude that the first alternative is better -- provided that the set of points x at which the first cdf is larger is sufficiently small.

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Technical Report UTEP-CS-14-37, May 2014

Protecting patient privacy while preserving medical information for research

Gang Xiang, Jason O'Rawe, Vladik Kreinovich, Janos Hajagos, and Scott Ferson

Published in *Proceedings of the 6th International Workshop on Reliable
Engineering Computing REC'2014*, Chicago, Illinois, May 25-28, 2014, pp. 281-293.

Patient health records possess a great deal of information that would be useful in medical research, but access to these data is impossible or severely limited because of the private nature of most personal health records. Anonymization strategies, to be effective, must usually go much further than simply omitting explicit identifiers because even statistics computed from groups of records can often be leveraged by hackers to re-identify individuals. Methods of balancing the informativeness of data for research with the information loss required to minimize disclosure risk are needed before these private data can be widely released to researchers who can use them to improve medical knowledge and public health. We are developing an integrated software system that provides solutions for anonymizing data based on interval generalization, controlling data utility, and performing statistical analyses and making inferences using interval statistics.

Technical Report UTEP-CS-14-36, May 2014

Updated version UTEP-CS-14-36c, January 2015

How Design Quality Improves with Increasing Computational Abilities: General Formulas and Case Study of Aircraft Fuel Efficiency

Joe Lorkowski, Olga Kosheleva, Vladik Kreinovich, and Sergei Soloviev

Short version published in *Proceedings of the International Symposium on
Management Engineering ISME'2014*, Kitakyushu, Japan, July 27-30,
2014, pp. 33-35; full paper published in *Journal of Advanced Computational
Intelligence and Intelligent Informatics (JACIII)*,
2015, Vol. 19, No. 5, pp. 581-584.

It is known that the problems of optimal design are NP-hard -- meaning that, in general, a feasible algorithm can only produce close-to-optimal designs. The more computations we perform, the better design we can produce. In this paper, we theoretically derive quantitative formulas describing how the design qualities improves with the increasing computational abilities. We then empirically confirm the resulting theoretical formula by applying it to the problem of aircraft fuel efficiency.

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Technical Report UTEP-CS-14-35, April 2014

Updated version UTEP-CS-14-35a, December 2015

In category of sets and relations, it is possible to describe functions in purely category terms

Vladik Kreinovich, Martine Ceberio, and Quentin Brefort

Published in *Eurasian Mathematical Journal*, 2015, Vol. 6,
No. 2, pp. 90-94.

We prove that in the category of sets and relations, it is possible to describe functions in purely category terms.

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Technical Report UTEP-CS-14-34, April 2014

Updated version UTEP-CS-14-34a, June 2014

How to Estimate Relative Spatial Resolution of Different Maps or Images of the Same Area?

Christian Servin, Aaron Velasco, and Vladik Kreinovich

Published in *Proceedings of IEEE International Conference on
Systems, Man, and Cybernetics SMC'2014*, San Diego, California,
October 5-8, 2014, pp. 3507-3511.

In this paper, we describe how to estimate relative spatial resolution of different maps or images of the same area under uncertainty. We consider probabilistic and fuzzy approaches and we show that both approaches lead to the same estimates -- which makes us more confident that this joint result is reasonable.

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Technical Report UTEP-CS-14-33, April 2014

Updated version UTEP-CS-14-33a, June 2014

How to Assign Weights to Different Factors in Vulnerability Analysis: Towards a Justification of a Heuristic Technique

Beverly Rivera, Irbis Gallegos, and Vladik Kreinovich

Published in *Mathematical Structures and Modeling*, 2014,
Vol. 30, pp. 87-98.

The main objective of vulnerability analysis is to select the alternative which is the least vulnerable. To make this selection, we must describe the vulnerability of each alternative by a single number -- then we will select the alternative with the smallest value of this vulnerability index. Usually, there are many aspects of vulnerability: vulnerability of a certain asset to a storm, to a terrorist attack, to hackers' attack, etc. For each aspect, we can usually gauge the corresponding vulnerability, the difficulty is how to combine these partial vulnerabilities into a single weighted value. In our previous research, we proposed an empirical idea of selecting the weights proportionally to the number of times the corresponding aspect is mentioned in the corresponding standards and requirements. This idea was shown to lead to reasonable results. In this paper, we provide a possible theoretical explanation for this empirically successful idea.

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Technical Report UTEP-CS-14-32, April 2014

Updated version UTEP-CS-14-32a, June 2014

r-Bounded Fuzzy Measures are Equivalent to epsilon-Possibility Measures

Karen Richart, Olga Kosheleva, and Vladik Kreinovich

Published in *Proceedings of IEEE International Conference on
Systems, Man, and Cybernetics SMC'2014*, San Diego, California,
October 5-8, 2014, pp. 1229-1234.

Traditional probabilistic description of uncertainty is based on additive probability measures. To describe non-probabilistic uncertainty, it is therefore reasonable to consider non-additive measures. An important class of non-additive measures are possibility measures, for which m(A union B) = max(m(A), m(B)). In this paper, we show that possibility measures are, in some sense, universal approximators: for every epsilon > 0, every non-additive measure which satisfies a certain reasonable boundedness property is equivalent to a measure which is epsilon-close to a possibility measure.

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Technical Report UTEP-CS-14-31, April 2014

Writing Self-testing Java Classes with SelfTest

Yoonsik Cheon

This document provides a tutorial introduction to Java annotations called SelfTest. The SelfTest annotations allow one to annotate Java classes with test data, and the SelfTest annotation processor generates executable JUnit test classes from annotated Java classes by translating test cases to executable JUnit tests. The SelfTest annotations not only automate unit testing of Java classes significantly but also provides a step toward writing self-testing Java classes by embedding test data in source code for both compile and runtime processing.

Technical Report UTEP-CS-14-30, March 2014

Detailed version UTEP-CS-14-30a, September 2014

Fuzzy intervals as foundation of metrological support for computations with inaccurate data

Konstantin K. Semenov, Gennady N. Solopchenko, and Vladik Kreinovich

Short version published in *Proceedings of the
International Conference on Advanced Mathematical and
Computational Tools in Metrology and Testing AMTCM'2014*,
St. Petersburg, Russia, September 9-12, 2014, Paper 088; full paper
published in: Franco Pavese,
Wolfram Bremser, Anna Chunovkina, Nicolas Fisher, and Alistair B.
Forbes (eds.),
*Advanced Mathematical and Computational Tools in Metrology and
Testing AMTCM'X*, World Scientific, Singapore, 2015, pp. 340-349.

In this paper, we discuss the possibility of using the formalism of fuzzy intervals as a basis for computational metrology. We consider advantages of using fuzzy intervals instead of the traditional intervals as a characteristic of uncertainty of the results of computations with inaccurate data.

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Technical Report UTEP-CS-14-29, March 2014

Deep mathematical results are the ones that connect seemingly unrelated areas: towards a formal proof of Gian-Carlo Rota's thesis

Olga Kosheleva and Vladik Kreinovich

Published in *Applied Mathematical Sciences*, 2014,
Vol. 8, No. 48, pp. 2391-2396.

When is a mathematical result deep? At first glance, the answer to this question is subjective: what is deep for one mathematician may not sound that deep for another. A renowned mathematician Gian-Carlo Rota expressed an opinion that the notion of deepness is more objective that we may think: namely, that a mathematical statement is deep if and only if it connects two seemingly unrelated areas of mathematics. In this paper, we formalize this thesis, and show that in this formalization, Gian Carlo Rota's thesis becomes a provable mathematical result.

Technical Report UTEP-CS-14-28, March 2014

Roadmap for Graduating Students with Expertise in the Analysis and Development of Secure Cyber-Systems

Ann Q. Gates, Salamah Salamah, and Luc Longpre

Modern society is intensely and irreversibly dependent on software systems of extraordinary size and complexity. This includes software systems in domain areas such as defense, energy, communication, transportation, and manufacturing. Due to the rapid expansion and reliance on the global Internet for day-to-day functions of individuals, organizations, governments, and industry around the world, cyber-security has emerged as an essential component of computing curricula. To address regional and national needs, the Computer Science Department has defined a roadmap for educating and preparing students who have expertise in the analysis and development of secure cyber-systems. Toward that vision, the department has set the following goals: (1) to increase the number of qualified students who complete a Secure Cyber-Systems track at the undergraduate or graduate levels at UTEP; (2) to graduate students who can enter the workforce with the ability to transfer state-of-the-art cybersecurity techniques and approaches into practice; (3) to place students in positions that utilize their knowledge and capabilities in cybersecurity. This technical report describes a set of objectives for each goal and the activities that will be implemented to achieve the objectives.

Technical Report UTEP-CS-14-27, March 2014

Fuzzy Logic Ideas Can Help in Explaining Kahneman and Tversky's Empirical Decision Weights

Joe Lorkowski and Vladik Kreinovich

Published in *Proceedings of the 4th World
Conference on Soft Computing*, Berkeley, California,
May 25-27, 2014; detailed version published in: Lotfi Zadeh
et al. (Eds.), *Recent Developments and New
Direction in Soft-Computing Foundations and Applications*,
Springer Verlag, 2016, pp. 89-98.

Analyzing how people actually make decisions, the Nobelist Daniel Kahneman and his co-author Amos Tversky found out that instead of maximizing the expected gain, people maximize a weighted gain, with weights determined by the corresponding probabilities. The corresponding empirical weights can be explained qualitatively, but quantitatively, these weights remains largely unexplained. In this paper, we show that with a surprisingly high accuracy, these weights can be explained by fuzzy logic ideas.

Technical Report UTEP-CS-14-26, March 2014

Updated version UTEP-CS-14-26a, May 2014

Wiener's Conjecture About Transformation Groups Helps Predict Which Fuzzy Techniques Work Better

Francisco Zapata, Olga Kosheleva, and Vladik Kreinovich

Published in *Proceedings of the 2014 Annual Conference of the North American
Fuzzy Information Processing Society NAFIPS'2014*,
Boston, Massachusetts, June 24-26, 2014.

Often, application success only comes when we select specific fuzzy techniques (t-norm, membership function, etc.) -- and in different applications, different techniques are the best. How to find the best technique? Exhaustive search of all techniques is not an option: there are too many of them. We need to come up with a narrow class of promising techniques, so that trying them all is realistic. In this paper, we show that such a narrowing can be obtained from transformation groups techniques motivated by N. Wiener's conjecture -- which was, in its turn, motivated by observations about human vision.

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Technical Report UTEP-CS-14-25, March 2014

Updated version UTEP-CS-14-25a, May 2014

Towards Efficient Algorithms for Approximating a Fuzzy Relation by Fuzzy Rules: Case When "And"- and "Or"-Operation are Distributive

Christian Servin and Vladik Kreinovich

Published in *Proceedings of the 2014 Annual Conference of the North American
Fuzzy Information Processing Society NAFIPS'2014*,
Boston, Massachusetts, June 24-26, 2014.

s A generic fuzzy relation often requires too many parameters to represent -- especially when we have a relation between many different quantities x1, ..., xn. There is, however, a class of relations which require much fewer parameters to describe - namely, relations which come from fuzzy rules. It is therefore reasonable to approximate a given relation by fuzzy rules. In this paper, we explain how this can be done in an important case when "and"- and "or"-operation are distributive -- and we also explain why this case is important.

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Technical Report UTEP-CS-14-24, March 2014

Updated version UTEP-CS-14-24a, May 2014

Interval and Symmetry Approaches to Uncertainty -- Pioneered by Wiener -- Helps Explain Many Seemingly Irrational Human Behaviors: A Case Study

Joe Lorkowski and Vladik Kreinovich

Published in *Proceedings of the 2014 Annual Conference of the North American
Fuzzy Information Processing Society NAFIPS'2014*,
Boston, Massachusetts, June 24-26, 2014.

It has been observed that in many cases, when we present a user with three selections od different price (and, correspondingly, different quality), then the user selects the middle selection. This empirical fact -- known as a compromise effect -- seems to contradicts common sense. Indeed, when a rational decision-maker selects one of the two alternatives, and then we add an additional option, then the user will either keep the previous selection or switch to a new option, but he/she will not select a previously rejected option. However, this is exactly what happens under the compromise effect. If we present the user with three options a < a' < a'', then, according to the compromise effect, the user will select the middle option a', meaning that between a' and a'', the user will select a'. However, if instead we present the user with three options a' < a'' < a''', then, according to the same compromise effect, the use will select a previously rejected option a''. In this paper, we show that this seemingly irrational behavior actually makes sense: it can be explained by an application of a symmetry approach, an approach whose application to uncertainty was pioneered by N. Wiener (together with interval approach to uncertainty).

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Technical Report UTEP-CS-14-23, March 2014

For each mathematical statement, only finitely many of its generalizations are useful: a formal proof of E. Bishop's idea

Olga Kosheleva and Vladik Kreinovich

Published in *International Mathematical Forum*, 2014,
Vol. 9, No. 16, pp. 763-766.

Generalization is one of the main mathematical activities. Some generalizations turn out to be useful for working mathematics, while many other generalizations have so far been not very useful. E. Bishop believed that most fruitless-so-far generalizations are hopeless, that every mathematical statement has only a few useful generalizations. In this paper, we show that, under a natural definition of the notion of useful generalization, Bishop's belief can be proven -- moreover, it turns out that for each mathematical statement, only finitely many of its generalizations are useful.

Technical Report UTEP-CS-14-22, March 2014

From Interval-Valued Probabilities to Interval-Valued Possibilities: Case Studies of Interval Computation under Constraints

Luis C. Gutierrez, Martine Ceberio, Vladik Kreinovich, Rebekah L. Gruver, Mariana Pena, Matthew J. Rister, Abraham Saldana, John Vasquez, Janelle Ybarra, and Salem Benferhat

Published in *Proceedings of the 6th International Workshop on Reliable
Engineering Computing REC'2014*, Chicago, Illinois, May 25-28, 2014, pp. 77-95.

In many engineering situations, we need to make decisions under
uncertainty. In some cases, we know the probabilities p_{i}
of different situations i; these probabilities should add up to 1.
In other cases, we only have expert estimates of the degree of
possibility μ_{i}i of different situations; in
accordance with the possibility theories, the largest of these
degrees should be equal to 1.

In practice, we often only know these degrees p_{i} and
μ_{i}i with uncertainty. Usually, we know the upper
bound and the lower bound on each of these values. In other words,
instead of the exact value of each degree, we only know the
*interval* of its possible values, so we need to process such
interval-valued degrees.

Before we start processing, it is important to find out which values from these intervals are actually possible. For example, if only have two alternatives, and the probability of the first one is 0.5, then -- even if the original interval for the second probability is wide -- the only possible value of the second probability is 0.5. Once the intervals are narrowed down to possible values, we need to compute the range of possible values of the corresponding characteristics (mean, variance, conditional probabilities and possibilities, etc.). For each such characteristic, first, we need to come up with an algorithm for computing its range.

In many engineering applications, we have a large amount of data to process, and many relevant decisions need to be made in real time. Because of this, it is important to make sure that the algorithms for computing the desired ranges are as fast as possible.

We present expressions for narrowing interval-valued probabilities and possibilities and for computing characteristics such as mean, conditional probabilities, and conditional possibilities. A straightforward computation of these expressions would take time which is quadratic in the number of inputs n. We show that in many cases, linear-time algorithms are possible -- and that no algorithm for computing these expressions can be faster than linear-time.

Technical Report UTEP-CS-14-21, March 2014

Decision Making under Interval Uncertainty: What Can and What Cannot Be Computed in Linear Time and in Real Time

Olga Kosheleva and Vladik Kreinovich

Published in *Proceedings of the 6th International Workshop on Reliable
Engineering Computing REC'2014*, Chicago, Illinois, May 25-28, 2014, pp. 116-124.

In engineering, we constantly need to make decisions: which design to select, which parameters to select for this design, etc.

The traditional approach to decision making is based on the assumption that we know all possible consequences of each alternative, and we know the probability of each such consequence. Under this assumption, we can describe a rational decision-making process: to each possible consequence, we assign a numerical values called its utility, and we select the alternative for which the expected value of the utility is the largest.

An important advantage of this approach is that it can be performed
in *real time:* if after we made a decision, a new alternative
appears, we do not need to repeat the whole analysis again: all we
need to do is compare the new alternative with the previously
selected ones.

In the past, when we used the same procedures year after year, we accumulated a lot of data about the consequences of different decisions -- based from which we could estimate the desired probabilities. Nowadays, with new technologies, new materials constantly emerging, we do not have such detailed information about the consequences of these new technologies. As a result, we often only have partial information about the corresponding probabilities. Different possible probability values result in different values of expected utility. Hence, for each alternative, instead of a single value of expected utility, we have a range (interval) of possible values. We need to make a decision under such interval uncertainty.

In this paper, we describe when we can make decisions under interval uncertainty in linear time and in real time -- and when we cannot.

Technical Report UTEP-CS-14-20, March 2014

Extending OCL to Better Express UML Qualified Associations

Alla Dove, Aditi Barua and Yoonsik Cheon

A qualified association in the Unified Modeling Language (UML) is an association that allows one to restrict the objects referred in an association using a key called a qualifier. A qualified association can appear in a constraint written in the Object Constraint Language (OCL) to specify a precise UML model. However, the OCL notation fails to provide appropriate support for expressing certain types of constraints written using qualified associations. In this paper we first describe a deficiency of OCL in expressing qualified associations and then propose a small extension to OCL to make it more expressive. The key idea of our extension is to view a qualified association as a map and provides a language construct to manipulate it as a first class entity in OCL. For this, we also extend the OCL standard library to introduce a wide range of map-specific collection operations. Our extension makes OCL not only more expressive but also amenable to a more direct translation to programming languages for various implementation uses of OCL constraints.

Technical Report UTEP-CS-14-19, March 2014

Revised version UTEP-CS-14-19a, July 2014

Construction of Shear Wave Models by Applying Multi-Objective Optimization to Multiple Geophysical Data Sets

Lennox Thompson, Aaron A. Velasco, and Vladik Kreinovich

Published in: Gerard Olivar Tost and Olga Vasilieva (eds.),
*Analysis, Modelling, Optimization, and Numerical Techniques*,
Springer Verlag, Berlin, Heidelberg, 2015, pp. 151-172.

For this work, our main purpose is to obtain a better understanding of the Earth’s tectonic processes in the Texas region, which requires us to analyze the Earth structure. We expand on a constrained optimization approach for a joint inversion least-squares (LSQ) algorithm to characterize a one-dimensional Earth's structure of Texas with the use of multiple geophysical data sets. We employed a joint inversion scheme using multiple geophysical datasets for the sole purpose of obtaining a three-dimensional velocity structure of Texas in order to identify an ancient rift system within Texas. In particular, we use data from the USArray, which is part of the EarthScope experiment, a 15-year program to place a dense network of permanent and portable seismographs across the continental United States. Utilizing the USArray data has provided us with the ability to image the crust and upper mantle structure of Texas. We simultaneously inverted multiple datasets from USArray data, to help us to better obtain an estimate of the true Earth structure model. We prove through numerical and experimental testing that our Multi-Objective Optimization (MOP) scheme performs inversion in a more robust, and flexible matter than traditional inversion approaches.

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Updated version in pdf

Technical Report UTEP-CS-14-18, March 2014

Detailed version UTEP-CS-14-18a, September 2014

Updated version UTEP-CS-14-18b, January 2015

Inverse problems in theory and practice of measurements and metrology

Konstantin K. Semenov, Gennadi N. Solopchenko, and Vladik Kreinovich

Short version published in *Proceedings of the
International Conference on Advanced Mathematical and
Computational Tools in Metrology and Testing AMTCM'2014*,
St. Petersburg, Russia, September 9-12, 2014, Paper 050; full paper
published in: Franco Pavese,
Wolfram Bremser, Anna Chunovkina, Nicolas Fisher, and Alistair B.
Forbes (eds.),
*Advanced Mathematical and Computational Tools in Metrology and
Testing AMTCM'X*, World Scientific, Singapore, 2015, pp. 330-339.

In this paper, we consider the role of inverse problems in metrology. We describe general methods of solving inverse problems which are useful in measurements practice. We also discuss how to modify these methods in situations in which there is a need for real-time data processing.

Short version UTEP-CS-14-18 in pdf

Detailed version UTEP-CS-14-18a in pdf

Updated version UTEP-CS-14-18b in pdf

Technical Report UTEP-CS-14-17, March 2014

Simpler-to-Describe Cases are Often More Difficult to Prove: A Possible Explanation

Olga Kosheleva and Vladik Kreinovich

Published in *International Mathematical Forum*, 2014,
Vol. 9, No. 16, pp. 767-772.

In many areas of mathematics, simpler-to-describe cases are often more difficult to prove. In this paper, we provide examples of such phenomena (Bieberbach's Conjecture, Poincar\'e Conjecture, Fermat's Last Theorem), and we provide a possible explanation for this empirical fact.

Technical Report UTEP-CS-14-16, March 2014

Updated version UTEP-CS-14-16a, March 204

Logic of Scientific Discovery: How Physical Induction Affects What Is Computable

Vladik Kreinovich and Olga Kosheleva

Published in *Proceedings of the
International Interdisciplinary Conference
Philosophy, Mathematics, Linguistics: Aspects of Interaction 2014
PhML'2014*, St. Petersburg, Russia, April 21-25, 2014, pp. 116-127.

Most of our knowledge about a physical world comes from physical induction: if a hypothesis is confirmed by a sufficient number of observations, we conclude that this hypothesis is universally true. We show that a natural formalization of this property affects what is computable when processing measurement and observation results, and we explain how this formalization is related to Kolmogorov complexity and randomness. We also consider computational consequences of an alternative idea also coming form physics: that no physical law is absolutely true, that every physical law will sooner or later need to be corrected. It turns out that this alternative approach enables us to use measurement results go beyond what is usually computable.

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Technical Report UTEP-CS-14-15, February 2014

Updated version UTEP-CS-14-15a, June 2014

Dealing with Uncertainties in Computing: from Probabilistic and Interval Uncertainty to Combination of Different Types of Uncertainty

Vladik Kreinovich

Published in: Gerard Olivar Tost and Olga Vasilieva (eds.),
*Analysis, Modelling, Optimization, and Numerical Techniques*,
Springer Verlag, Berlin, Heidelberg, 2015, pp. 309-326.

To predict values of future quantities, we apply algorithms to the current and past measurement results. Because of the measurement errors and model inaccuracy, the resulting estimates are, in general, different from the desired values of the corresponding quantities. There exist methods for estimating this difference, but these methods have been mainly developed for the two extreme cases: the case when we know the exact probability distributions of all the measurement errors and the interval case, when we only know the bounds on the measurement errors. In practice, we often have some partial information about the probability distributions which goes beyond these bounds. In this paper, we show how the existing methods of estimating uncertainty can be extended to this generic case.

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Technical Report UTEP-CS-14-14, February 2014

Interleaving Enhances Learning: A Possible Geometric Explanation

Octavio Lerma, Olga Kosheleva, and Vladik Kreinovich

Published in *Geombinatorics*, 2015, Vol. 24, No. 3,
pp. 135-139.

In the traditional approach to learning, if we want students to learn how to solve different types of problems, we first teach them how to solve problems of the first type, then how to solve problems of the second type, etc. It turns out that we can speed up learning if we interleave problems of different types. In particular, it has bene empirically shown that interleaving problems of four different types leads to a double speed-up. In this paper, we provide a possible geometric explanation for this empirical fact.

Technical Report UTEP-CS-14-13, February 2014

Range Estimation under Constraints is Computable Unless There Is a Discontinuity

Martine Ceberio, Olga Kosheleva, and Vladik Kreinovich

Published in *Proceedings of the Seventh International
Workshop on Constraints Programming and Decision Making
CoProd'2014*, Wuerzburg, Germany, September 21, 2014; detailed version
published in: Martine Ceberio and Vladik Kreinovich (eds.),
*Constraint Programming and Decision Making: Theory and
Applications*, Springer Verlag, Berlin, Heidelberg, 2018,
pp. 39-44.

One of the main problems of interval computations is computing the range of a given function over given intervals. It is known that there is a general algorithm for computing the range of computable functions over computable intervals. However, if we take into account that often in practice, not all possible combinations of the inputs are possible (i.e., that there are constraints), then it becomes impossible to have an algorithm which would always compute this range. In this paper, we explain that the main reason why range estimation under constraints is not always computable is that constraints may introduce discontinuity -- and all computable functions are continuous. Specifically, we show that if we restrict ourselves to computably continuous constraints, the problem of range estimation under constraints remains computable.

Technical Report UTEP-CS-14-12, February 2014

Fuzzy, Intuitionistic Fuzzy, What Next?

Vladik Kreinovich and Bui Cong Cuong

Published in: in: Plamen Angelov and Sotir Sotirov (ed.),
*Imprecision and Uncertainty in Information Representation and
Processing*, Springer Verlag, 2016, pp. 3-14.

In the 1980s, Krassimir Atanassov proposed an important generalization of fuzzy sets, fuzzy logic, and fuzzy techniques -- intuitionistic fuzzy approach, which provides a more accurate description of expert knowledge. In this paper, we describe a natural way how the main ideas behind the intuitionistic fuzzy approach can be expanded even further, towards an even more accurate description of experts' knowledge.

Technical Report UTEP-CS-14-11, February 2014

Fitts's Law: Towards a Geometric Explanation

Olga Kosheleva, Vladik Kreinovich, and Octavio Lerma

Published in *Geombinatorics*, 2014, Vol. 24, No. 2,
pp. 78-83.

In designing human-computer interfaces, designers use an empirical Fitts's Law, according to which the average time T of accessing an icon of size w at a distance d from the center of the screen is proportional to the logarithm of the ratio w/d. There exist explanations for this law, but these explanations have gaps. In this paper, we show that these gaps can be explained if we analyze this problem from the geometric viewpoint. Thus, we get a geometric explanation of the Fitts's Law.

Technical Report UTEP-CS-14-10, January 2014

Updated version UTEP-CS-14-10a, February 2014

How to Compare Different Range Estimations: A Symmetry-Based Approach

O. Kosheleva and V. Kreinovich

Published in *Proceedings of the American Society of Civil Engineers
(ASCE) Second International Conference on Vulnerability
and Risk Analysis and Management ICVRAM'2014 and Sixth
International Symposium on Uncertainty Modelling and
Analysis ISUMA'2014*, Liverpool, UK, July 13-16, 2014,
pp. 340-349.

How to compare different range estimators for multivariate functions under uncertainty? To answer this question, we analyze which utility functions can be used for this task. Specifically, we: (1) introduce various invariance assumptions, (2) describe the class of all utility functions which satisfy these assumptions, and (3) show how the resulting utility functions can be used to compare different range estimators.

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Technical Report UTEP-CS-14-09, January 2014

Updated version UTEP-CS-14-09a, February 2014

Towards Efficient Ways of Estimating Failure Probability of Mechanical Structures Under Interval Uncertainty

M. Beer, M. De Angelis, and V. Kreinovich

Published in *Proceedings of the American Society of Civil
Engineers
(ASCE) Second International Conference on Vulnerability
and Risk Analysis and Management ICVRAM'2014 and Sixth
International Symposium on Uncertainty Modelling and
Analysis ISUMA'2014*, Liverpool, UK, July 13-16, 2014,
pp. 320-329.

Whether the structure is stable depends on the values of the parameters
which describe the structure and its environment. Usually, we know the *limit function*
describing stability: a structure is stable if and only if the value of this
function is positive. If we also know the probability distribution on the set
of all possible combinations of parameters, then we can estimate the failure probability P.

In practice, we often know that the probability distribution belongs to
the known family of distributions (e.g., normal), but we only know the
approximate values of the parameters characterizing the actual distribution.
Similarly, we know the family of possible limit functions, but we have only
approximate estimates P_{i} of the parameters p_{i}
corresponding to the actual limit
function. In many such situations, we know the accuracy of the corresponding
approximations, i.e., we know an upper bound D_{i} for which
|P_{i} − p_{i}| is smaller than or equal to
D_{i}. In this case, the only information that we have about
the actual (unknown) values of the corresponding parameters p_{i} is that
p_{i} is in the interval [P_{i} − D_{i}, P_{i} + D_{i}].
Different values p_{i} from the corresponding intervals lead, in general, to different values
of the failure probability P. So, under such interval uncertainty, it is desirable to
find the range [P] of possible values of failure probability. In this paper,
we describe efficient algorithms for computing this range.

We also show how to take into account the *model inaccuracy*, i.e., the fact
that the finite-parametric models of the distribution and of the limit
function provide only an approximate descriptions of the actual ones.

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Technical Report UTEP-CS-14-08, January 2014

From Global to Local Constraints: A Constructive Version of Bloch's Principle

Martine Ceberio, Olga Kosheleva, and Vladik Kreinovich

Published in *Proceedings of the Seventh International
Workshop on Constraints Programming and Decision Making
CoProd'2014*, Wuerzburg, Germany, September 21, 2014;
detailed version published in: Martine Ceberio and
Vladik Kreinovich (eds.),
*Constraint Programming and Decision Making: Theory and
Applications*, Springer Verlag, Berlin, Heidelberg, 2018,
pp. 27-32.

Generalizing several results from complex analysis, A. Bloch formulated an informal principle -- that for every global implication there is a stronger local implication. This principle has been formalized for complex analysis, but is has been successfully used in other areas as well. In this paper, we propose a new formalization of Bloch's Principle, and we show that in general, the corresponding localized version can be obtained algorithmically.

Technical Report UTEP-CS-14-07, January 2014

Updated version UTEP-CS-14-07a, March 2014

Towards Decision Making under Interval, Set-Valued, Fuzzy, and Z-Number Uncertainty: A Fair Price Approach

Joe Lorkowski, Rafik Aliev, and Vladik Kreinovich

Published in *Proceedings of the IEEE World Congress on
Computational Intelligence*, Beijing, China, July 6-11,
2014.

In this paper, we explore one of the possible ways to make decisions under uncertainty: namely, we explain how to define a fair price for a participation in such a decision, and then select an alternative for which the corresponding fair price is the largest. This idea is explained on the examples of interval uncertainty, set-valued, fuzzy, and Z-number uncertainty.

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Technical Report UTEP-CS-14-06, January 2014

A Feasible Algorithm for Checking n-Scissors Congruence of Polyhedra in R

Olga Kosheleva and Vladik Kreinovich

Published in *Geombinatorics*, 2015, Vol. 25, No. 2, pp. 70-75.

While in R^{2}, every two polygons of the same area are
*scissors congruent* (i.e., they can be both decomposed
into the same finite number of pair-wise congruent
polygonal pieces), in R^{3}, there are polyhedra P and P' of
the same volume which are not scissors-congruent. It is
therefore necessary, given two polyhedra, to check whether they
are scissors-congruent (and if yes -- to find the corresponding decompositions).
It is known that while there are algorithms for performing this
checking-and-finding task, no such algorithm can be
feasible -- their worst-case computation time grows (at least) exponentially,
so even for reasonable size inputs, the computation time exceeds the lifetime
of the Universe. It is therefore desirable to
find cases when feasible algorithms are possible.

In this paper, we show that for each dimension d, a feasible algorithm is possible
if we fix some integer n and look for *n-scissors-congruence* in R^{d} -- i.e., for possibility to
represent P and P' as a union of n (of fewer) *simplexes*.

Technical Report UTEP-CS-14-05, January 2014

Why Injecting Fine Dust into a Tornado Is More Promising Than Injecting Coarse Dust: A Geometric Explanation

Octavio Lerma, Olga Kosheleva, and Vladik Kreinovich

Published in *Geombinatorics*, 2016, Vol. 25, No. 3, pp.
118-122.

One of the promising ways to tame a tornado is to inject dust into it. Somewhat counter-intuitively, injecting coarse dust only makes the tornado stronger, while injecting fine dust can indeed help in the taming. This difference has been explained by a mathematical analysis of the corresponding equations, but (in contrast to the usual physics practice) this mathematical analysis has not yet been accompanied by a simple qualitative physical explanation. We show that such a simple explanation can be obtained if we analyze the problem of taming tornados from the geometric viewpoint.

Technical Report UTEP-CS-14-04, January 2014

Diversity Is Beneficial for a Research Group: One More Quantitative Argument

Komsan Suriya, Tatcha Sudtasan, Tonghui Wang, Octavio Lerma, and Vladik Kreinovich

Published in *Journal of Uncertain Systems*, 2015, Vol. 9, No. 2,
pp. 144-147.

In this paper, we propose a natural model describing competition between two research groups of the same average research strength. The analysis of this model enables us to conclude that a more diverse group has an advantage: namely, the more diverse the group, the higher the average quality of its publications.

Technical Report UTEP-CS-14-03, January 2014

A Simple Geometric Model Provides a Possible Quantitative Explanation of the Advantages of Immediate Feedback in Student Learning

Octavio Lerma, Olga Kosheleva, and Vladik Kreinovich

Published in *Geombinatorics*, 2015, Vol. 25, No. 1, pp. 22-29.

Calculus is a known bottleneck for many students studying science and engineering. Various techniques have been developed to enhance the students' success. A recent study published in the Notices of American Mathematical Society showed that only one factor determines the success of a technique: the presence of immediate feedback. On average, students who receive immediate feedback learn twice faster than students who are taught in a more traditional way, with a serious feedback only once or twice a semester (after a test).

The very fact that immediate feedback is helpful is not surprising: it helps the student clear misconceptions and avoid the wrong paths. However, the fact that different techniques involving feedback lead to practically the same learning speed-up is intriguing. To explain this speed-up, we provide a simplified first-order description of a learning process in simple geometric terms. We show that already in this first approximation, the geometric description leads to the observed two-fold speed-up in learning.

Technical Report UTEP-CS-14-02, January 2014

Updated version UTEP-XS-14-02a, March 2014

How to Understand Connections Based on Big Data: From Cliques to Flexible Granules

Ali Jalal-Kamali, M. Shahriar Hossain, and Vladik Kreinovich

Published in: Shyi-Ming Chen and Witold Pedrycz (eds.),
*Information Granularity, Big Data, and Computational Intelligence*,
Springer Verlag, Cham, Switzerland, 2015, pp. 63-87.

One of the main objectives of science and engineering is to predict the future state of the world -- and to come up with actions which will lead to the most favorable outcome. To be able to do that, we need to have a quantitative model describing how the values of the desired quantities change -- and for that, we need to know which factors influence this change. Usually, these factors are selected by using traditional statistical techniques, but with the current drastic increase in the amount of available data -- known as the advent of {\it big data} -- the traditional techniques are no longer feasible. A successful semi-heuristic method has been proposed to detect true connections in the presence of big data. However, this method has its limitations. The first limitation is that this method is heuristic -- its main justifications are common sense and the fact that in several practical problems, this method was reasonably successful. The second limitation is that this heuristic method is based on using "crisp" granules (clusters), while in reality, the corresponding granules are flexible ("fuzzy"). In this paper, we explain how the known semi-heuristic method can be justified in statistical terms, and we also show how the ideas behind this justification enable us to improve the known method by taking granule flexibility into account.

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Technical Report UTEP-CS-14-01, January 2014

Zipf's Law and 7 Plus Minus 2 Principle Lead to a Possible Explanation of Daniel's Law

Olga Kosheleva and Vladik Kreinovich

Published in *International Mathematical Forum*, 2014, Vol. 9,
No. 8, pp. 391-396.

In 1961, D. R. Daniel observed that the success of a company is usually determined by three to six major factors. This observation has led to many successful management ideas, but they leave one puzzled: why three to six? why not two or seven? In this paper, we provide a possible explanation to this puzzle; namely, we show that these numbers of factors can be derived from Zipf's Law and from the 7 plus minus 2 principle.