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Full-Text Articles in Mathematics
For Discrete-Time Linear Dynamical Systems Under Interval Uncertainty, Predicting Two Moments Ahead Is Np-Hard, Luc Jaulin, Olga Kosheleva, Vladik Kreinovich
For Discrete-Time Linear Dynamical Systems Under Interval Uncertainty, Predicting Two Moments Ahead Is Np-Hard, Luc Jaulin, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In the first approximation, when changes are small, most real-world systems are described by linear dynamical equations. If we know the initial state of the system, and we know its dynamics, then we can, in principle, predict the system's state many moments ahead. In practice, however, we usually know both the initial state and the coefficients of the system's dynamics with some uncertainty. Frequently, we encounter interval uncertainty, when for each parameter, we only know its range, but we have no information about the probability of different values from this range. In such situations, we want to know the range …
What To Do If An Inflexible Tolerance Problem Has No Solutions: Probabilistic Justification Of Piegat's Semi-Heuristic Idea, Olga Kosheleva, Vladik Kreinovich
What To Do If An Inflexible Tolerance Problem Has No Solutions: Probabilistic Justification Of Piegat's Semi-Heuristic Idea, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In many practical situations, it is desirable to select the control parameters x1, ..., xn in such a way that the resulting quantities y1, ..., ym of the system lie within desired ranges. In such situations, we usually know the general formulas describing the dependence of yi on xj, but the coefficients of these formulas are usually only known with interval uncertainty. In such a situation, we want to find the tuples for which all yi's are in the desired intervals for all possible tuples of coefficients. But what if no such parameters are possible? Since we cannot guarantee the …
How To Make Ai More Reliable, Olga Kosheleva, Vladik Kreinovich
How To Make Ai More Reliable, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
One of the reasons why the results of the current AI methods (especially deep-learning-based methods) are not absolutely reliable is that, in contrast to more traditional data processing techniques which are based on solid mathematical and statistical foundations, modern AI techniques use a lot of semi-heuristic methods. These methods have been, in many cases, empirically successful, but the absence of solid justification makes us less certain that these methods will work in other cases as well. To make AI more reliable, it is therefore necessary to provide mathematical foundations for the current semi-heuristic techniques. In this paper, we show that …
Why Magenta Is Not A Real Color, And How It Is Related To Fuzzy Control And Quantum Computing, Victor L. Timchenko, Yuriy P. Kondratenko, Olga Kosheleva, Vladik Kreinovich
Why Magenta Is Not A Real Color, And How It Is Related To Fuzzy Control And Quantum Computing, Victor L. Timchenko, Yuriy P. Kondratenko, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
It is well known that every color can be represented as a combination of three basic colors: red, green, and blue. In particular, we can get several colors by combining two of the basic colors. Interestingly, while a combination of two neighboring colors leads to a color that corresponds to a certain frequency, the combination of two non-neighboring colors -- red and blue -- leads to magenta, a color that does not correspond to any frequency. In this paper, we provide a simple explanation for this phenomenon, and we also show that a similar phenomenon happens in two other areas …
How To Propagate Uncertainty Via Ai Algorithms, Olga Kosheleva, Vladik Kreinovich
How To Propagate Uncertainty Via Ai Algorithms, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Any data processing starts with measurement results. Measurement results are never absolutely accurate. Because of this measurement uncertainty, the results of processing measurement results are, in general, somewhat different from what we would have obtained if we knew the exact values of the measured quantities. To make a decision based on the result of data processing, we need to know how accurate is this result, i.e., we need to propagate the measurement uncertainty through the data processing algorithm. There are many techniques for uncertainty propagation. Usually, they involve applying the same data processing algorithm several times to appropriately modified data. …
Why Empirical Membership Functions Are Well-Approximated By Piecewise Quadratic Functions: Theoretical Explanation For Empirical Formulas Of Novak's Fuzzy Natural Logic, Olga Kosheleva, Vladik Kreinovich
Why Empirical Membership Functions Are Well-Approximated By Piecewise Quadratic Functions: Theoretical Explanation For Empirical Formulas Of Novak's Fuzzy Natural Logic, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Empirical analysis shows that membership functions describing expert opinions have a shape that is well described by a smooth combination of two quadratic segments. In this paper, we provide a theoretical explanation for this empirical phenomenon.
Why Is Grade Distribution Often Bimodal? Why Individualized Teaching Adds Two Sigmas To The Average Grade? And How Are These Facts Related?, Christian Servin, Olga Kosheleva, Vladik Kreinovich
Why Is Grade Distribution Often Bimodal? Why Individualized Teaching Adds Two Sigmas To The Average Grade? And How Are These Facts Related?, Christian Servin, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
To make education more effective, to better use emerging technologies in education, we need to better understand the education process, to gain insights on this process. How can we check whether a new idea is indeed a useful insight? A natural criterion is that the new idea should explain some previously-difficult-to-explain empirical phenomenon. Since one of the main advantages of emerging educational technologies -- such as AI -- is the possibility of individualized education, a natural phenomenon to explain is the fact -- discovered by Benjamin Bloom -- that individualization adds two sigmas to the average grade. In this paper, …
How To Make A Neural Network Learn From A Small Number Of Examples -- And Learn Fast: An Idea, Chitta Baral, Vladik Kreinovich
How To Make A Neural Network Learn From A Small Number Of Examples -- And Learn Fast: An Idea, Chitta Baral, Vladik Kreinovich
Departmental Technical Reports (CS)
Current deep learning techniques have led to spectacular results, but they still have limitations. One of them is that, in contrast to humans who can learn from a few examples and learn fast, modern deep learning techniques require a large amount of data to learn, and they take a long time to train. In this paper, we show that neural networks do have a potential to learn from a small number of examples -- and learn fast. We speculate that the corresponding idea may already be implicitly implemented in Large Language Models -- which may partially explain their (somewhat mysterious) …
How Can We Explain Empirical Formulas For Shrinkage Cracking Of Cement-Stabilized Pavement Layers, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich
How Can We Explain Empirical Formulas For Shrinkage Cracking Of Cement-Stabilized Pavement Layers, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich
Departmental Technical Reports (CS)
In pavement construction, one of the frequent defects is shrinkage cracking of the cement-stabilized pavement layer. To minimize this defect, it is important to be able to predict how this cracking depends on the quantities describing the pavement layer and the corresponding environment. Cracking is usually described by two parameters: the average width of the crack and the crack spacing. Empirical analysis shows that the dependence of the width on all related quantities is described by a power law. Power laws are ubiquitous in physics, they describe a frequent case when the dependence is scale-invariant -- i.e., does not change …
Topics In The Study Of The Pragmatic Functions Of Phonetic Reduction In Dialog, Nigel G. Ward, Carlos A. Ortega
Topics In The Study Of The Pragmatic Functions Of Phonetic Reduction In Dialog, Nigel G. Ward, Carlos A. Ortega
Departmental Technical Reports (CS)
Reduced articulatory precision is common in speech, but for dialog its acoustic properties and pragmatic functions have been little studied. We here try to remedy this gap. This technical report contains content that was omitted from the journal article (Ward et. al, submitted). Specifically, we here report 1) lessons learned about annotating for perceived reduction, 2) the finding that, unlike in read speech, the correlates of reduction in dialog include high pitch, wide pitch range, and intensity, and 3) a baseline model for predicting reduction in dialog, using simple acoustic/prosodic features, that achieves correlations with human perceptions of 0.24 for …
Towards An Optimal Design: What Can We Recommend To Elon Musk?, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich, Hung T. Nguyen
Towards An Optimal Design: What Can We Recommend To Elon Musk?, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich, Hung T. Nguyen
Departmental Technical Reports (CS)
Elon Musk's successful "move fast and break things" strategy is based on the fact that in many cases, we do not need to satisfy all usual constraints to be successful. By sequentially trying smaller number of constraints, he finds the smallest number of constraints that are still needed to succeed -- and using this smaller number of constrains leads to a much cheaper (and thus, more practical) design. In this strategy, Musk relies on his intuition -- which, as all intuitions, sometimes works and sometimes doesn't. To replace this intuition, we propose an algorithm that minimizes the worst-case cost of …
Shall We Place More Advanced Students In A Separate Class?, Shahnaz Shahbazova, Olga Kosheleva, Vladik Kreinovich
Shall We Place More Advanced Students In A Separate Class?, Shahnaz Shahbazova, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In every class, we have students who are more advanced and students who are more behind. From this viewpoint, it seems reasonable to place more advanced students in a separate class. This should help advanced students progress faster, and it should help other students as well, since the teachers in the remaining class can better attend to their needs. However, empirically, this does not work: when we form a separate class, the overall amount of gained knowledge decreases. In this paper, we provide a possible explanation for this seemingly counterintuitive phenomenon.
How Difficult Is It To Comprehend A Program That Has Significant Repetitions: Fuzzy-Related Explanations Of Empirical Results, Christian Servin, Olga Kosheleva, Vladik Kreinovich
How Difficult Is It To Comprehend A Program That Has Significant Repetitions: Fuzzy-Related Explanations Of Empirical Results, Christian Servin, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In teaching computing and in gauging the programmers' productivity, it is important to property estimate how much time it will take to comprehend a program. There are techniques for estimating this time, but these techniques do not take into account that some program segments are similar, and this similarity decreases the time needed to comprehend the second segment. Recently, experiments were performed to describe this decrease. These experiments found an empirical formula for the corresponding decrease. In this paper, we use fuzzy-related ideas to provide commonsense-based theoretical explanation for this empirical formula.
Mcfadden's Discrete Choice And Softmax Under Interval (And Other) Uncertainty: Revisited, Bartlomiej Jacek Kubica, Olga Kosheleva, Vladik Kreinovich
Mcfadden's Discrete Choice And Softmax Under Interval (And Other) Uncertainty: Revisited, Bartlomiej Jacek Kubica, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Studies of how people actually make decisions have led to an empirical formula that predicts the probability of different decisions based on the utilities of different alternatives. This formula is known as McFadden's formula, after a Nobel prize winning economist who discovered it. A similar formula -- known as softmax -- describes the probability that the classification predicted by a deep neural network is correct, based on the neural network's degrees of confidence in the object belonging to each class. In practice, we usually do not know the exact values of the utilities -- or of the degrees of confidence. …
Why Bernstein Polynomials: Yet Another Explanation, Olga Kosheleva, Vladik Kreinovich
Why Bernstein Polynomials: Yet Another Explanation, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In many computational situations -- in particular, in computations under interval or fuzzy uncertainty -- it is convenient to approximate a function by a polynomial. Usually, a polynomial is represented by coefficients at its monomials. However, in many cases, it turns out more efficient to represent a general polynomial by using a different basis -- of so-called Bernstein polynomials. In this paper, we provide a new explanation for the computational efficiency of this basis.
Somewhat Surprisingly, (Subjective) Fuzzy Technique Can Help To Better Combine Measurement Results And Expert Estimates Into A Model With Guaranteed Accuracy: Digital Twins And Beyond, Niklas Winnewisser, Michael Beer, Olga Kosheleva, Vladik Kreinovich
Somewhat Surprisingly, (Subjective) Fuzzy Technique Can Help To Better Combine Measurement Results And Expert Estimates Into A Model With Guaranteed Accuracy: Digital Twins And Beyond, Niklas Winnewisser, Michael Beer, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
To understand how different factors and different control strategies will affect a system -- be it a plant, an airplane, etc. -- it is desirable to form an accurate digital model of this system. Such models are known as digital twins. To make a digital twin as accurate as possible, it is desirable to incorporate all available knowledge of the system into this model. In many cases, a significant part of this knowledge comes in terms of expert statements, statements that are often formulated by using imprecise ("fuzzy") words from natural language such as "small", "very possible", etc. To translate …
How To Gauge Inequality And Fairness: A Complete Description Of All Decomposable Versions Of Theil Index, Saeid Tizpaz-Niari, Olga Kosheleva, Vladik Kreinovich
How To Gauge Inequality And Fairness: A Complete Description Of All Decomposable Versions Of Theil Index, Saeid Tizpaz-Niari, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In general, in statistics, the most widely used way to describe the difference between different elements of a sample if by using standard deviation. This characteristic has a nice property of being decomposable: e.g., to compute the mean and standard deviation of the income overall the whole US, it is sufficient to compute the number of people, mean, and standard deviation over each state; this state-by-state information is sufficient to uniquely reconstruct the overall standard deviation. However, e.g., for gauging income inequality, standard deviation is not very adequate: it provides too much weight to outliers like billionaires, and thus, does …
Update From Aristotle To Newton, From Sets To Fuzzy Sets, And From Sigmoid To Relu: What Do All These Transitions Have In Common?, Christian Servin, Olga Kosheleva, Vladik Kreinovich
Update From Aristotle To Newton, From Sets To Fuzzy Sets, And From Sigmoid To Relu: What Do All These Transitions Have In Common?, Christian Servin, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In this paper, we show that there is a -- somewhat unexpected -- common trend behind several seemingly unrelated historic transitions: from Aristotelian physics to modern (Newton's) approach, from crisp sets (such as intervals) to fuzzy sets, and from traditional neural networks, with close-to-step-function sigmoid activation functions to modern successful deep neural networks that use a completely different ReLU activation function. In all these cases, the main idea of the corresponding transition can be explained, in mathematical terms, as going from the first order to second order differential equations.
How To Make A Decision Under Interval Uncertainty If We Do Not Know The Utility Function, Jeffrey Escamilla, Vladik Kreinovich
How To Make A Decision Under Interval Uncertainty If We Do Not Know The Utility Function, Jeffrey Escamilla, Vladik Kreinovich
Departmental Technical Reports (CS)
Decision theory describes how to make decisions, in particular, how to make decisions under interval uncertainty. However, this theory's recommendations assume that we know the utility function -- a function that describes the decision maker's preferences. Sometimes, we can make a recommendation even when we do not know the utility function. In this paper, we provide a complete description of all such cases.
Paradox Of Causality And Paradoxes Of Set Theory, Alondra Baquier, Bradley Beltran, Gabriel Miki-Silva, Olga Kosheleva, Vladik Kreinovich
Paradox Of Causality And Paradoxes Of Set Theory, Alondra Baquier, Bradley Beltran, Gabriel Miki-Silva, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Logical paradoxes show that human reasoning is not always fully captured by the traditional 2-valued logic, that this logic's extensions -- such as multi-valued logics -- are needed. Because of this, the study of paradoxes is important for research on multi-valued logics. In this paper, we focus on paradoxes of set theory. Specifically, we show their analogy with the known paradox of causality, and we use this analogy to come up with similar set-theoretic paradoxes.
Number Representation With Varying Number Of Bits, Anuradha Choudhury, Md Ahsanul Haque, Saeefa Rubaiyet Nowmi, Ahmed Ann Noor Ryen, Sabrina Saika, Vladik Kreinovich
Number Representation With Varying Number Of Bits, Anuradha Choudhury, Md Ahsanul Haque, Saeefa Rubaiyet Nowmi, Ahmed Ann Noor Ryen, Sabrina Saika, Vladik Kreinovich
Departmental Technical Reports (CS)
In a computer, usually, all real numbers are stored by using the same number of bits: usually, 8 bytes, i.e., 64 bits. This amount of bits enables us to represent numbers with high accuracy -- up to 19 decimal digits. However, in most cases -- whether we process measurement results or whether we process expert-generated membership degrees -- we do not need that accuracy, so most bits are wasted. To save space, it is therefore reasonable to consider representations with varying number of bits. This would save space used for representing numbers themselves, but we would also need to store …
Data Fusion Is More Complex Than Data Processing: A Proof, Robert Alvarez, Salvador Ruiz, Martine Ceberio, Vladik Kreinovich
Data Fusion Is More Complex Than Data Processing: A Proof, Robert Alvarez, Salvador Ruiz, Martine Ceberio, Vladik Kreinovich
Departmental Technical Reports (CS)
Empirical data shows that, in general, data fusion takes more computation time than data processing. In this paper, we provide a proof that data fusion is indeed more complex than data processing.
How To Fairly Allocate Safety Benefits Of Self-Driving Cars, Fernando Munoz, Christian Servin, Vladik Kreinovich
How To Fairly Allocate Safety Benefits Of Self-Driving Cars, Fernando Munoz, Christian Servin, Vladik Kreinovich
Departmental Technical Reports (CS)
In this paper, we describe how to fairly allocated safety benefits of self-driving cars between drivers and pedestrians -- so as to minimize the overall harm.
Using Known Relation Between Quantities To Make Measurements More Accurate And More Reliable, Niklas Winnewisser, Felix Mett, Michael Beer, Olga Kosheleva, Vladik Kreinovich
Using Known Relation Between Quantities To Make Measurements More Accurate And More Reliable, Niklas Winnewisser, Felix Mett, Michael Beer, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Most of our knowledge comes, ultimately, from measurements and from processing measurement results. In this, metrology is very valuable: it teaches us how to gauge the accuracy of the measurement results and of the results of data processing, and how to calibrate the measuring instruments so as to reach the maximum accuracy. However, traditional metrology mostly concentrates on individual measurements. In practice, often, there are also relations between the current values of different quantities. For example, there is usually an known upper bound on the difference between the values of the same quantity at close moments of time or at …
Why Pavement Cracks Are Mostly Longitudinal, Sometimes Transversal, And Rarely Of Other Directions: A Geometric Explanation, Edgar Daniel Rodriguez Velasquez, Olga Kosheleva, Vladik Kreinovich
Why Pavement Cracks Are Mostly Longitudinal, Sometimes Transversal, And Rarely Of Other Directions: A Geometric Explanation, Edgar Daniel Rodriguez Velasquez, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In time, pavements deteriorate, and need maintenance. One of the most typical pavement faults are cracks. Empirically, the most frequent cracks are longitudinal, i.e., following the direction of the road; less frequent are transversal cracks, which are orthogonal to the direction of the road. Sometimes, there are cracks in different directions, but such cracks are much rarer. In this paper, we show that simple geometric analysis and fundamental physical ideas can explain these observed relative frequencies.
Why Linear And Sigmoid Last Layers Work Better In Classification, Lehel Dénes-Fazakas, Lásló Szilágyi, Vladik Kreinovich
Why Linear And Sigmoid Last Layers Work Better In Classification, Lehel Dénes-Fazakas, Lásló Szilágyi, Vladik Kreinovich
Departmental Technical Reports (CS)
Usually, when a deep neural network is used to classify objects, its last layer computes the softmax. Our empirical results show we can improve the classification results if instead, we have linear or sigmoid last layer. In this paper, we provide an explanation for this empirical phenomenon.
Why Two Fish Follow Each Other But Three Fish Form A School: A Symmetry-Based Explanation, Shahnaz Shahbazova, Olga Kosheleva, Vladik Kreinovich
Why Two Fish Follow Each Other But Three Fish Form A School: A Symmetry-Based Explanation, Shahnaz Shahbazova, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Recent experiments with fish has shown an unexpected strange behavior: when two fish of the same species are placed in an aquarium, they start following each other, while when three fish are placed there, they form (approximately) an equilateral triangle, and move in the direction (approximately) orthogonal to this triangle. In this paper, we use natural symmetries -- such as rotations, shifts, and permutation of fish -- to show that this observed behavior is actually optimal. This behavior is not just optimal with respect to one specific optimality criterion, it is optimal with respect to any optimality criterion -- as …
Fuzzy Ideas Explain Fechner Law And Help Detect Relation Between Objects In Video, Olga Kosheleva, Vladik Kreinovich, Ahnaf Farhan
Fuzzy Ideas Explain Fechner Law And Help Detect Relation Between Objects In Video, Olga Kosheleva, Vladik Kreinovich, Ahnaf Farhan
Departmental Technical Reports (CS)
How to find relation between objects in a video? If two objects are closely related -- e.g., a computer and it mouse -- then they almost always appear together, and thus, their numbers of occurrences are close. However, simply computing the differences between numbers of occurrences is not a good idea: objects with 100 and 110 occurrences are most probably related, but objects with 1 and 5 occurrences probably not, although 5 − 1 is smaller than 110 − 100. A natural idea is, instead, to compute the difference between re-scaled numbers of occurrences, for an appropriate nonlinear re-scaling. In …
There Is Still Plenty Of Room At The Bottom: Feynman's Vision Of Quantum Computing 65 Years Later, Alexis Lupo, Vladik Kreinovich, Victor L. Timchenko, Yuriy P. Kondratenko
There Is Still Plenty Of Room At The Bottom: Feynman's Vision Of Quantum Computing 65 Years Later, Alexis Lupo, Vladik Kreinovich, Victor L. Timchenko, Yuriy P. Kondratenko
Departmental Technical Reports (CS)
In 1959, Nobelist Richard Feynman gave a talk titled "There's plenty of room at the bottom", in which he emphasized that, to drastically speed up computations, we need to make computer components much smaller -- all the way to the size of molecules, atoms, and even elementary particles. At this level, physics is no longer described by deterministic Newton's mechanics, it is described by probabilistic quantum laws. Because of this, computer designers started thinking how to design a reliable computer based on non-deterministic elements -- and this thinking eventually led to the modern ideas and algorithms of quantum computing. So, …
From Quantifying And Propagating Uncertainty To Quantifying And Propagating Both Uncertainty And Reliability: Practice-Motivated Approach To Measurement Planning And Data Processing, Niklas R. Winnewisser, Vladik Kreinovich, Olga Kosheleva
From Quantifying And Propagating Uncertainty To Quantifying And Propagating Both Uncertainty And Reliability: Practice-Motivated Approach To Measurement Planning And Data Processing, Niklas R. Winnewisser, Vladik Kreinovich, Olga Kosheleva
Departmental Technical Reports (CS)
When we process data, it is important to take into account that data comes with uncertainty. There exist techniques for quantifying uncertainty and propagating this uncertainty through the data processing algorithms. However, most of these techniques do not take into account that in real world, measuring instruments are not 100% reliable -- they sometimes malfunction and produce values which are far off from the measured values of the corresponding quantities. How can we take into account both uncertainty and reliability? In this paper, we consider several possible scenarios, and we show, for each scenario, what is the natural way to …