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Full-Text Articles in Mathematics
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.
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.
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.
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.
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 …
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 …
Every Feasibly Computable Reals-To-Reals Function Is Feasibly Uniformly Continuous, Olga Kosheleva, Vladik Kreinovich
Every Feasibly Computable Reals-To-Reals Function Is Feasibly Uniformly Continuous, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
It is known that every computable function is continuous; moreover, it is computably continuous in the sense that for every ε > 0, we can compute δ > 0 such that δ-close inputs lead to ε-close outputs. It is also known that not all functions which are, in principle, computable, can actually be computed: indeed, the computation sometimes requires more time than the lifetime of the Universe. A natural question is thus: can the above known result about computable continuity of computable functions be extended to the case when we limit ourselves to feasible computations? In this paper, we prove that this …
From Normal Distribution To What? How To Best Describe Distributions With Known Skewness, Olga Kosheleva, Vladik Kreinovich
From Normal Distribution To What? How To Best Describe Distributions With Known Skewness, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In many practical situations, we only have partial information about the probability distribution -- e.g., all we know is its few moments. In such situations, it is desirable to select one of the possible probability distributions. A natural way to select a distribution from a given class of distributions is the maximum entropy approach. For the case when we know the first two moments, this approach selects the normal distribution. However, when we also know the third central moment -- corresponding to skewness -- a direct application of this approach does not work. Instead, practitioners use several heuristic techniques, techniques …
Every Relu-Based Neural Network Can Be Described By A System Of Takagi-Sugeno Fuzzy Rules: A Theorem, Barnabas Bede, Olga Kosheleva, Vladik Kreinovich
Every Relu-Based Neural Network Can Be Described By A System Of Takagi-Sugeno Fuzzy Rules: A Theorem, Barnabas Bede, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
While modern deep-learning neural networks are very successful, sometimes they make mistakes, and since their results are "black boxes" -- no explanation is provided -- it is difficult to determine which recommendations are erroneous. It is therefore desirable to make the resulting computations explainable, i.e., to describe their results by using commonsense rules. In this paper, we use "fuzzy" techniques -- techniques developed by Lotfi Zadeh to deal with commonsense rules formulated by using imprecise ("fuzzy") words from natural language -- to show that such a rule-based representation is always possible. Our result does not yet provide the desired explainability, …
Smooth Non-Additive Integrals And Measures And Their Potential Applications, Olga Kosheleva, Vladik Kreinovich
Smooth Non-Additive Integrals And Measures And Their Potential Applications, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In this paper, we explain why non-additive integrals and measures are needed, how non-additive integrals and measures are related, how to use them in decision making, and how they can help in fundamental physics. These four topics are covered, correspondingly, in Sections 2-5 of this paper.
Why Sigmoid Transformation Helps Incorporate Logic Into Deep Learning: A Theoretical Explanation, Chitta Baral, Vladik Kreinovich
Why Sigmoid Transformation Helps Incorporate Logic Into Deep Learning: A Theoretical Explanation, Chitta Baral, Vladik Kreinovich
Departmental Technical Reports (CS)
Traditional neural networks start from the data, they cannot easily handle prior knowledge -- this is one of the reasons why they often take very long to train. It is desirable to incorporate prior knowledge into deep learning. For the case when this knowledge consists of propositional statements, a successful way to incorporate this knowledge was proposed in a recent paper by van Krieken et al. That paper uses the fact that a neural network does not directly return a truth value, it returns a real value -- in effect, the degree of confidence in the corresponding statement -- from …
If We Add Axiom Of Choice To Constructive Analysis, We Get Classical Arithmetic: An Exercise In Reverse Constructive Mathematics, Olga Kosheleva, Vladik Kreinovich
If We Add Axiom Of Choice To Constructive Analysis, We Get Classical Arithmetic: An Exercise In Reverse Constructive Mathematics, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
A recent paper in Bulletin of Symbolic Logic reminded that the Axiom of Choice is, in general, false in constructive analysis. This result is an immediate consequence of a theorem -- first proved by Tseytin -- that every computable function is continuous. In this paper, we strengthen the result about the Axiom of Choice by proving that this axiom is as non-constructive as possible: namely, that if we add this axiom to constructive analysis, then we get full classical arithmetic.
When Is A Single "And"-Condition Enough?, Olga Kosheleva, Vladik Kreinovich
When Is A Single "And"-Condition Enough?, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In many practical situations, there are several possible decisions. Any general recommendation means specifying, for each possible decision, conditions under which this decision is recommended. In some cases, a single "and"-condition is sufficient: e.g., a condition under which a patient is recommended to take aspirin is that "the patient has a fever and the patient does not have stomach trouble". In other cases, conditions are more complicated. A natural question is: when is a single "and"-condition enough? In this paper, we provide an answer to this question.
From Type-2 Fuzzy To Type-2 Intervals And Type-2 Probabilities, Vladik Kreinovich, Olga Kosheleva, Luc Longpré
From Type-2 Fuzzy To Type-2 Intervals And Type-2 Probabilities, Vladik Kreinovich, Olga Kosheleva, Luc Longpré
Departmental Technical Reports (CS)
Our knowledge comes from observations, measurements, and expert opinions. Measurements and observations are never 100% accurate, there is always a difference between the measurement result and the actual value of the corresponding quantity. We gauge the resulting uncertainty either by an interval of possible values, or by a probability distribution on the set of possible values, or by a membership function that describes to what extent different values are possible. The information about uncertainty also comes either from measurements or from expert estimates and is, therefore, also uncertain. It is important to take such "type-2" uncertainty into account. This is …
Giant Footprints Of Buddha And Generalized Limits, Julio C. Urenda, Vladik Kreinovich
Giant Footprints Of Buddha And Generalized Limits, Julio C. Urenda, Vladik Kreinovich
Departmental Technical Reports (CS)
In many places in Asia, there are footprints claimed to be left by Buddha. Many of them are much larger than the usual size of human feet, up to 150 cm and more in length. In this paper, we provide a possible mathematical explanation for such unusual sizes.
How To Efficiently Propagate P-Box Uncertainty, Olga Kosheleva, Vladik Kreinovich
How To Efficiently Propagate P-Box Uncertainty, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In many practical situations, to get the desired estimate or prediction, we need to process existing data. This data usually comes from measurements, and measurements are never 100% accurate. Because we only know the input values with uncertainty, the results of processing this data also comes with uncertainty. To make an appropriate decision, we need to know how accurate is the resulting estimate, i.e., how the input uncertainty "propagates" through the data processing algorithm. In the ideal case, when we know the probability distribution of each measurement error, we can, in principle, use Monte-Carlo simulations to describe the uncertainty of …
Uncertainty Quantification For Results Of Ai-Based Data Processing: Towards More Feasible Algorithms, Christoph Q. Lauter, Martine Ceberio, Vladik Kreinovich, Olga Kosheleva
Uncertainty Quantification For Results Of Ai-Based Data Processing: Towards More Feasible Algorithms, Christoph Q. Lauter, Martine Ceberio, Vladik Kreinovich, Olga Kosheleva
Departmental Technical Reports (CS)
AI techniques have been actively and successfully used in data processing. This tendency started with fuzzy techniques, now neural network techniques are actively used. With each new technique comes the need for the corresponding uncertainty quantification (UQ). In principle, for both fuzzy and neural techniques, we can use the usual UQ methods -- however, these techniques often require an unrealistic amount of computation time. In this paper, we show that in both cases, we can use specific features of the corresponding techniques to drastically speed up the corresponding computations.
Usually, Either Left And Right Brains Are Equally Active Or Only One Of Them Is Active: First-Principles Explanation, Julio C. Urenda, Vladik Kreinovich
Usually, Either Left And Right Brains Are Equally Active Or Only One Of Them Is Active: First-Principles Explanation, Julio C. Urenda, Vladik Kreinovich
Departmental Technical Reports (CS)
It is known that in most practical situations, either both left and right brains are equally active, or only one of them is active. A recent paper showed that this empirical phenomenon can be explained by a realistic model of the brain effectiveness. In this paper, we show that this conclusion can be made without any specific assumptions about the brain, based on first principles.
Which Random-Set Representation Of A Fuzzy Set Is The Simplest?, Vladik Kreinovich, Olga Kosheleva, Hung T. Nguyen
Which Random-Set Representation Of A Fuzzy Set Is The Simplest?, Vladik Kreinovich, Olga Kosheleva, Hung T. Nguyen
Departmental Technical Reports (CS)
One of the ways to elicit membership degrees is by polling. For example, we ask a group of people how many believe that 30 C is hot. If 8 out of ten say that it is hot, we assign the degree 8/10 to the statement "30 C is hot". In precise mathematical terms, polling can be described via so-called random sets. It is known that every fuzzy set can be obtained this way, i.e., that every fuzzy set can be represented by an appropriate random set. Moreover, it is known that for many fuzzy sets, there are several different random-set …
Why Micro-Funding? Why Small Businesses Are Important? Analysis Based On First Principles, Hein D. Tran, Edwin Tomy George, Vladik Kreinovich
Why Micro-Funding? Why Small Businesses Are Important? Analysis Based On First Principles, Hein D. Tran, Edwin Tomy George, Vladik Kreinovich
Departmental Technical Reports (CS)
On the one hand, in economics, there is a well-known and well-studied economy of scale: when two smaller companies merge, it lowers their costs and thus, makes them more effective and therefore more competitive. At first glance, this advantage of big size would make economy dominated by big companies -- but in reality, small business remain a significant and important economic sector. Similarly, it is well known and well studied that research collaboration enhances researchers' productivity -- but still a significant portion of important results come from individual efforts. In several applications areas, there are area-specific explanations for this seemingly …
Local-Global Support For Earth Sciences: Economic Analysis, Uyen Hoang Pham, Aaron Velasco, Vladik Kreinovich
Local-Global Support For Earth Sciences: Economic Analysis, Uyen Hoang Pham, Aaron Velasco, Vladik Kreinovich
Departmental Technical Reports (CS)
Most funding for science comes from taxpayers. So, it is very important to be able to convince taxpayers that this funding is potentially beneficial for them. This task is easier in Earth sciences, e.g., in meteorology, where there are clear local benefits. The problem is that while many people support local studies focused on their region, they do not always have a good understanding of the fact that effective local benefits require also studying surrounding areas -- and what should be the optimal balance between local and (more) global studies. In this paper, on a (somewhat) simplified model of the …
Approximate Stochastic Dominance Revisited, Chon Van Le, Olga Kosheleva, Vladik Kreinovich
Approximate Stochastic Dominance Revisited, Chon Van Le, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
According to decision theory, in general, to recommend the best of possible actions, we need to know, for each possible action, the probabilities of different outcomes, and we also need to know the decision maker's utility function -- that describes his/her preferences. For some pairs of probability distributions, however, we can make such a recommendation without knowing the exact form of the utility function -- e.g., in financial applications, we only need to know that a larger amount is preferable to a smaller one. Such situations, when we can make decisions based only on the information about probabilities, are known …
Just-In-Accuracy: Mobile Approach To Uncertainty, Martine Ceberio, Christoph Q. Lauter, Vladik Kreinovich
Just-In-Accuracy: Mobile Approach To Uncertainty, Martine Ceberio, Christoph Q. Lauter, Vladik Kreinovich
Departmental Technical Reports (CS)
To make a mobile device last longer, we need to limit computations to a bare minimum. One way to do that, in complex control and decision making problems, is to limit precision with which we do computations, i.e., limit the number of bits in the numbers' representation. A problem is that often, we do not know with what precision should we do computations to get the desired accuracy of the result. What we propose is to first do computations with very low precision, then, based on these computations, estimate what precision is needed to achieve the given accuracy, and then …
How To Deal With Inconsistent Intervals: Utility-Based Approach Can Overcome The Limitations Of The Purely Probability-Based Approach, Kittawit Autchariyapanitkul, Tomoe Entani, Olga Kosheleva, Vladik Kreinovich
How To Deal With Inconsistent Intervals: Utility-Based Approach Can Overcome The Limitations Of The Purely Probability-Based Approach, Kittawit Autchariyapanitkul, Tomoe Entani, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In many application areas, we rely on experts to estimate the numerical values of some quantities. Experts can provide not only the estimates themselves, they can also estimate the accuracies of their estimates -- i.e., in effect, they provide an interval of possible values of the quantity of interest. To get a more accurate estimate, it is reasonable to ask several experts -- and to take the intersection of the resulting intervals. In some cases, however, experts overestimate the accuracy of their estimates, their intervals are too narrow -- so narrow that they are inconsistent: their intersection is empty. In …
How To Make Machine Learning Financial Recommendations More Fair: Theoretical Explanation, Tho M. Nguyen, Saeid Tizpaz-Niari, Vladik Kreinovich
How To Make Machine Learning Financial Recommendations More Fair: Theoretical Explanation, Tho M. Nguyen, Saeid Tizpaz-Niari, Vladik Kreinovich
Departmental Technical Reports (CS)
Machine learning has been actively and successfully used to make financial decisions. In general, these systems work reasonably well. However, in some cases, these systems show unexpected bias towards minority groups -- the bias that is sometime much larger than the bias in the data on which they were trained. A recent paper analyzed whether a proper selection of hyperparameters can decrease this bias. It turned out that while the selection of hyperparameters indeed affect the system's fairness, only a few of the hyperparameters lead to consistent improvement of fairness: the number of features used for training and the number …