Computer Engineering Commons^™

Our Reasoning Is Clearly Fuzzy, So Why Is Crisp Logic So Often Adequate?, Hung T. Nguyen, Berlin Wu, Vladik Kreinovich

Departmental Technical Reports (CS)

Our reasoning is clearly fuzzy, so why is crisp logic so often adequate? We explain this phenomenon by showing that in the presence of noise, an arbitrary continuous (e.g., fuzzy) system can be well described by its discrete analog. However, as the description gets more accurate, the continuous description becomes necessary.

How Design Quality Improves With Increasing Computational Abilities: General Formulas And Case Study Of Aircraft Fuel Efficiency, Joe Lorkowski, Olga Kosheleva, Vladik Kreinovich, Sergei Soloviev

Departmental Technical Reports (CS)

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.

Testing A Power Law Model Of Knowledge Propagation: Case Study Of The Out Of Eden Walk Project, Octavio Lerma, Leobardo Valera, Deana Pennington, Vladik Kreinovich

Departmental Technical Reports (CS)

To improve teaching and learning, it is important to understand how knowledge propagates. In general, when a new piece of knowledge is introduced, people start learning about it. Since the potential audience is limited, after some time, the number of new learners starts to decrease. Traditional models of knowledge propagation are based on differential equations; in these models, the number of new learners decreases exponentially with time. Recently, a new power law model for knowledge propagation was proposed. In this model, the number of learners decreases much slower, as a negative power of time. In this paper, we compare the …

Optimizing Pred(25) Is Np-Hard, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Usually, in data processing, to find the parameters of the models that best fits the data, people use the Least Squares method. One of the advantages of this method is that for linear models, it leads to an easy-to-solve system of linear equations. A limitation of this method is that even a single outlier can ruin the corresponding estimates; thus, more robust methods are needed. In particular, in software engineering, often, a more robust pred(25) method is used, in which we maximize the number of cases in which the model's prediction is within the 25% range of the observations. In …

Wiener-Process-Type Evasive Aircraft Actions Are Indeed Optimal Against Anti-Aircraft Guns: Wiener's Data Revisited, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

In his 1940s empirical study of evasive aircraft actions, N.~Wiener, the father of cybernetics, founds out that the pilot's actions follow a Wiener-type-process. In this paper, we explain this empirical result by showing that such evasive actions are indeed optimal against the 1940s anti-aircraft guns.

Fuzzy (And Interval) Techniques In The Age Of Big Data: An Overview With Applications To Environmental Science, Geosciences, Engineering, And Medicine, Vladik Kreinovich, Rujira Ouncharoen

Departmental Technical Reports (CS)

In some practical situations -- e.g., when treating a new illness -- we do not have enough data to make valid statistical conclusions. In such situations, it is necessary to use expert knowledge -- and thus, it is beneficial to use fuzzy techniques that were specifically designed to process such knowledge. At first glance, it may seem that in situations when we have large amounts of data, the relative importance of expert knowledge should decrease. However, somewhat surprisingly, it turns out that expert knowledge is still very useful in the current age of big data. In this paper, we explain …

Towards The Possibility Of Objective Interval Uncertainty In Physics. Ii, Luc Longpre, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Applications of interval computations usually assume that while we only know an interval containing the actual (unknown) value of a physical quantity, there is the exact value of this quantity, and that in principle, we can get more and more accurate estimates of this value. Physicists know, however, that, due to uncertainty principle, there are limitations on how accurately we can measure the values of physical quantities. One of the important principles of modern physics is operationalism -- that a physical theory should only use observable properties. This principle is behind most successes of the 20th century physics, starting with …

Minimax Portfolio Optimization Under Interval Uncertainty, Meng Yuan, Xu Lin, Junzo Watada, Vladik Kreinovich

Departmental Technical Reports (CS)

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 …

Constraint Approach To Multi-Objective Optimization, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we would like to maximize (or minimize) several different criteria, and it is not clear how much weight to assign to each of these criteria. Such situations are ubiquitous and thus, it is important to be able to solve the corresponding multi-objective optimization problems. There exist many heuristic methods for solving such problems. In this paper, we reformulate multi-objective optimization as a constraint satisfaction problem, and we show that this reformulation explains two widely use multi-objective optimization techniques: optimizing a weighted sum of the objective functions and optimizing the product of normalized values of these functions.

Interval Computations And Interval-Related Statistical Techniques: Estimating Uncertainty Of The Results Of Data Processing And Indirect Measurements, Vladik Kreinovich

Departmental Technical Reports (CS)

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.

Need For Data Processing Naturally Leads To Fuzzy Logic (And Neural Networks): Fuzzy Beyond Experts And Beyond Probabilities, Vladik Kreinovich, Hung T. Nguyen, Songsak Sriboonchitta

Departmental Technical Reports (CS)

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 …

Every Sue Function Is A Ratio Of Two Multi-Linear Functions, Joe Lorkowski, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

We prove that the function computed by each single-use expression is a ratio of two multi-linear functions.

When Can We Reduce Multi-Variable Range Estimation Problem To Two Fewer-Variable Problems?, Joe Lorkowski, Olga Kosheleva, Luc Longpre, Vladik Kreinovich

Departmental Technical Reports (CS)

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.

50 Years Of Fuzzy: From Discrete To Continuous To -- Where?, Vladik Kreinovich, Hung T. Nguyen, Olga Kosheleva, Rujira Ouncharoen

Departmental Technical Reports (CS)

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.

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, Vladik Kreinovich

Departmental Technical Reports (CS)

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.

Granularity Helps Explain Seemingly Irrational Features Of Human Decision Making, Joe Lorkowski, Vladik Kreinovich

Departmental Technical Reports (CS)

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.

Asymptotically Optimal Algorithm For Checking Whether A Given Vector Is A Solution To A Given Interval-Quantifier Linear System, Vladik Kreinovich

Departmental Technical Reports (CS)

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 …

Is The World Itself Fuzzy? Physical Arguments For -- And Unexpected Computational Consequences Of -- Zadeh's Vision, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

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 …

If We Take Into Account That Constraints Are Soft, Then Processing Constraints Becomes Algorithmically Solvable, Quentin Brefort, Luc Jaulin, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

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, …

Why Ricker Wavelets Are Successful In Processing Seismic Data: Towards A Theoretical Explanation, Afshin Gholamy, Vladik Kreinovich

Departmental Technical Reports (CS)

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.

Likert-Type Fuzzy Uncertainty From A Traditional Decision Making Viewpoint: How Symmetry Helps Explain Human Decision Making (Including Seemingly Irrational Behavior), Joe Lorkowski, Vladik Kreinovich

Departmental Technical Reports (CS)

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 …

A Catalog Of While Loop Specification Patterns, Aditi Barua, Yoonsik Cheon

Departmental Technical Reports (CS)

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.

Security Risk Assessment: Towards A Justification For The Security Risk Factor Table Model, Beverly Rivera, Francisco Zapata, Vladik Kreinovich

Departmental Technical Reports (CS)

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.

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, Vladik Kreinovich

Departmental Technical Reports (CS)

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. …

Towards A Formal Description Of Understandability (Causality, Pre-Requisites): From Prosorov's Phonocentric Topology To More General Interior (Closure) Structures, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

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 T₀).

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, …

From Numerical Probabilities To Linguistic Probabilities: A Theoretical Justification Of Empirical Granules Used In Risk Management, Beverly Rivera, Francisco Zapata, Vladik Kreinovich

Departmental Technical Reports (CS)

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.

Homotopy Techniques In Solving Systems Of Nonlinear Equations: A Theoretical Justification Of Convex Combinations, Nicholas Sun

Departmental Technical Reports (CS)

One of the techniques for solving systems of non-linear equations F₁(x₁,...,x_n) = 0, ..., F_n(x₁,...,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(x₁,...,x_n) = 0, and then gradually adjust this solution by solving intermediate systems of equation H_i(x₁,...,x_n) = 0 for an appropriate "transition" function H(x) = f(λ,F(x),G(x)). The success of this method depends on the …

Log-Periodic Power Law As A Predictor Of Catastrophic Events: A New Mathematical Justification, Vladik Kreinovich, Hung T. Nguyen, Songsak Sriboonchitta

Departmental Technical Reports (CS)

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 …

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, Rujira Ouncharoen

Departmental Technical Reports (CS)

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.

Increased Climate Variability Is More Visible Than Global Warming: A General System-Theory Explanation, Octavio Lerma, Craig Tweedie, Vladik Kreinovich

Departmental Technical Reports (CS)

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.