Published in: Van-Nam Huynh, Katsuhiro Honda, Bac H. Le, Masahiro Inuiguchi, and Hieu T. Huynh (eds.), Proceedings of the 11th International Symposium on Integrated Uncertainty in Knowledge Modelling and Decision Making IUKM 2025, Ho Chi Minh City, Vietnam, March 17-19, 2025, pp. 76-84.

For complex engineering systems, the usual way to estimate their reliability is to run simulations. If the resulting estimate does not satisfy the desired reliability level, we must replace some components with more reliable and again run simulations. This can take several iterations, so the required computation time often becomes unrealistically long. It is known that it is possible to speed up computations if components belong to a few types, and components of each type are identical. So, a natural idea to deal with the general case is to use the general granularity idea, i.e., to group components with similar reliability characteristics into a single cluster, and for each component from the cluster, use the same average reliability instead of the original (somewhat different) reliability characteristics. The accuracy of the resulting approximate estimate of the system's reliability depends on how exactly we divide the components into clusters. It is therefore desirable to select the clustering that leads to the most accurate reliability estimate. In this paper, we describe an algorithm for such optimal clustering.

Published in: Van-Nam Huynh, Katsuhiro Honda, Bac H. Le, Masahiro Inuiguchi, and Hieu T. Huynh (eds.), Proceedings of the 11th International Symposium on Integrated Uncertainty in Knowledge Modelling and Decision Making IUKM 2025, Ho Chi Minh City, Vietnam, March 17-19, 2025, pp. 85-97.

For a complex engineering system -- such as a city's street network -- it is important to predict how its functionality is decreased when some of these components break down, and, if repairs are needed and repairs budget is limited, which subset of the set of components should be repaired first to maximize the resulting functionality. For systems with a large number of components, the number of possible subsets is astronomical, we cannot try to simulate all these subsets. So, the natural idea is to approximate the actual dependence of functionality on the subset by a simple expression -- linear or quadratic -- and to use known algorithms for optimizing such approximate expressions. In this paper, we provide an algorithm for such an approximation, and we show that for linear approximations, the resulting expression is a generalization of Shapley value -- a techniques that is now successfully use to make machine-learning-based AI explainable. We also analyze how the Shapley value idea can be further improved.

Published in Asian Journal of Economics and Banking, 2025, DOI: 10.1108/AJEB-12-2024-0150

Purpose: When several participants, working together, gained some amount of money, what is the fair way to distribute this amount between them? This is the problem that the future Nobelist Lloyd Shapley was working on when he proposed what is now called the Shapley value -- a division uniquely determined by natural fairness assumptions. However, this solutions is not universal: it assumes that all participants are equal -- in particular, that they have equal productivity. In practice, people have different productivity levels, and these productivity levels can differ a lot: e.g., some software engineers are several times more productive than others. It is desirable to take this difference in productivity into account.

Design/methodology/approach: Shapley value is based on an axiomatic approach: it is uniquely determined by the appropriate fairness assumptions. To generalize Shapley value to the case of different productivity, we modified these assumptions appropriately, and analyzed what can be derived from these modified assumptions.

Findings: We prove that there is a unique division scheme that satisfies all the resulting assumptions. This scheme is thus a generalization of Shapley value to this more general and more realistic situation, when different participants have different productivity.

Originality/value: Both the formulation of the problem and the result are new. The resulting division scheme can be used to more adequately distribute the common gains -- by explicitly taking into account that different participants have, in general, different productivity.

To appear in Proceedings of the 16th European Symposium on Computational Intelligence and Mathematics ESCIM 2025, A Coruna, Spain, May 18-21, 2025.

Current AI systems are very successful, but they are not perfect, they need to be trained better. Training modern AI systems requires a tremendous amount of computations -- that already take a lot of time. To increase the number of computations, we need to make each computation step faster. It was shown that we can speed up computations if we apply an appropriate nonlinear transformation to all the values, and that logarithmic transformation leads to the fastest speedup. In this paper, we provide a theoretical explanation for this empirical success.

Published in Geombinatorics, 2025, Vol. 34, No. 4, pp. 156-161.

Earthquakes usually occur in the vicinity of fault lines. Until recently, geophysical analysis implied that the fault shape should not strongly affect the frequency of its earthquakes. However, recent statistical analysis has shown that faults whose shape is close to linear experience much fewer earthquakes than faults of more complex shape. Based on this empirical fact, researchers have adjusted the corresponding geophysical models, so the updated models do explain this newly discovered phenomenon. The experience of geophysics shows that the updated model will probably need to be updated again when new data appears. It is therefore desirable to come up with an explanation of the above phenomenon that does not depend on the specifics of the underlying physical model. In this paper, we provide such an explanation -- it is based only on the corresponding geometric symmetries and on the general physical ideas related to symmetries and their violations.

In several locations, geologists have observed the presence of two differently oriented rock masses, one horizonal (or almost horizontal) and the other somewhat inclined; this phenomenon is known as angular unconformity. Based on the detailed analysis of geophysical processes, geologists conclude that usually, horizontal rock masses are much newer. This is known as the law of original horizontality. From the fundamental viewpoint, it is desirable to take into account that geophysics is a developing science, its models get modified and adjusted as time progresses. It is therefore desirable to come up with an explanation of this phenomenon that would be maximally independent on any specifics of a geophysical model. In this paper, we show that the law of original horizontality can be derived by using only the general geometric symmetry ideas.

To appear in Philosophical Transactions of the Royal Society A: Mathematical, Physical, and Engineering Sciences, DOI 10.1098/rsta.2023.0290

From the physics viewpoint, energy is the ability to perform work. To estimate how much work we can perform, physicists developed several formalisms. For example, for the fields, once we know the Lagrangian, we can find the energy density and, by integrating it, estimate the overall energy of the field. Usually, this adequately describe how much work this field can perform. However, there is an exception -- gravitational field in General Relativity. The known formalism to compute its energy density leads to 0 -- and by integrating this 0, we get a counterintuitive conclusion that the overall energy of the gravitational field is 0 -- while hydroelectric power stations that produce a significant portion of world's energy show that gravity {\it can} perform a lot of work and thus, has non-pzero energy. The usual solution to this puzzle is that for gravity, energy is not localized. In this paper, we show: (1) that non-locality of energy can be explained already on the Newtonian level, (2) that the discrepancy between energy as ability to perform work and energy as described by the Lagrangian-based formalism is ubiquitous even in the Newtonian case, and (3) that there may be a possible positive side to this non-locality: it may lead to faster computations.

To appear in: A. I. Shevchenko and Yuriy P. Kondratenko (eds.), Artificial Intelligence: Achievements and Recent Developments, River Publishers, Denmark, 2024, to appear.

At present, the most successful AI technique is deep learning -- the use of neural networks that consist of multiple layers. Interestingly, it is well known that neural networks with two data processing layers are sufficient -- in the sense that they can approximate any function with any given accuracy. Because of this, until reasonably recently, researchers and practitioners used such networks. However, recently it turned out, somewhat unexpectedly, that using three or more data processing layers -- i.e., using what is called deep learning -- makes the neural networks much more efficient. In this paper, on numerous examples from AI and from beyond AI, we show that this is a general phenomenon: two is enough but three or more is better. In many examples, there is a specific explanation for this phenomenon. However, the fact that this phenomenon is universal makes us conjecture that there is a general explanation for this phenomenon -- and we provide a possible explanation.

In many practically useful numerical computations, training-and-then-using a neural network turned out to be a much faster alternative than running the original computations. When we applied a similar idea to take into account interval uncertainty, we encountered two unexpected results: (1) that while for numerical computations, it is usually better to represent an interval by its midpoint and half-width, for neural networks, it is more efficient to represent an interval by its endpoints, and (2) that while usually, it is better to train a neural network on the whole data processing algorithm, in our problems, it turned out to be more efficient to train several subnetworks on subtasks and then combine their results. In this paper, we provide a theoretical explanation for these unexpected results.

To appear in Geombinatorics

In this paper, we describe three applications of geometric reasoning to important practical problems ranging from micro- to macro-level. Specifically, we use geometric reasoning to explain why metastasis is mostly caused by elongated cancer cell, why curiosity in fish is strongly correlated with body shape, and why ring-shaped fractures appear in Antarctica.

To appear in: Nguyen Hoang Phuong, Nguyen Thi Huyen Chau, and Vladik Kreinovich (eds.), Explainable Artificial Intelligence and Other Soft Computing Techniques: Biomedical and Related Applications, Springer, Cham, Switzerland.

In many practical situations, a reasonable conjecture is that, e.g., the dependence of some quantity on the spatial location is continuous, with an appropriate bounds on the difference between the values at nearby points. If we knew the exact values of the corresponding quantity, checking this conjecture would be very straightforward. In reality, however, measurement results are only approximations to the actual values. In this paper, we show how to check continuity based on the approximate measurement results.

To appear in: Nguyen Hoang Phuong, Nguyen Thi Huyen Chau, and Vladik Kreinovich (eds.), Explainable Artificial Intelligence and Other Soft Computing Techniques: Biomedical and Related Applications, Springer, Cham, Switzerland.

In the usual deep neural network, weights are adjusted during training, but the activation function remains the same. Lately, it was experimentally shown that if, instead of using the same activation function always, we train the activation functions as well, we get a much better results -- i.e., for the networks with the same number of parameters, we get a much better accuracy. Such networks are called Kolmogorov-Arnold networks. In this paper, we provide a general explanation of why these new networks work so well.

To appear in: Nguyen Hoang Phuong, Nguyen Thi Huyen Chau, and Vladik Kreinovich (eds.), Explainable Artificial Intelligence and Other Soft Computing Techniques: Biomedical and Related Applications, Springer, Cham, Switzerland.

When two events are independent, the probability that both events occur is equal to the product p1 * p2 of the probabilities of each of these events. The probability that at least one of these events will occur is equal to p1 + p2 − p1 * p2. In both cases, we have a commutative associative polynomial operation. A natural question is: how can we describe all possible operations of this type? These operations are described in this paper.

To appear in: Nguyen Hoang Phuong, Nguyen Thi Huyen Chau, and Vladik Kreinovich (eds.), Explainable Artificial Intelligence and Other Soft Computing Techniques: Biomedical and Related Applications, Springer, Cham, Switzerland.

Interdisciplinary research is very important in modern science. However, such a research is not easy, it often needs support and help. Resources that can be used for such a support are limited, so we need to decide which of many possible collaborations we should support. In this paper, we provide a natural simple model of collaboration effectiveness. Based on this model, we conclude that we should support collaborations for which the vector product of the participants' knowledge vectors attains the largest values.

To appear in: Nguyen Ngoc Thach, Nguyen Duc Trung, Doan Thanh Ha, and Vladik Kreinovich, Artificial Intelligence and Machine Learning for Econometrics: Applications and Regulation (and Related Topics), Springer, Cham, Switzerland.

Often, to make an appropriate decision, people try three scenarios: the worst case, the most realistic case, and the best case. This three-scenarios approach often leads to reasonable decisions. A natural question is: why worst case and best case? These extreme cases mean that all numerous independent random factors work in the same direction: either are all stacked for or are all stacked against. Such stacking of random factors is highly improbable. So, at first glance, it would be more beneficial to use more realistic scenarios than the worst case and the best case. However, empirically, decisions based on the worst-case and the best-case scenarios work well -- better than other three-scenarios alternatives. In this paper, we provide a theoretical explanation for this empirical phenomenon.

To appear in: Nguyen Hoang Phuong, Nguyen Thi Huyen Chau, and Vladik Kreinovich (eds.), Explainable Artificial Intelligence and Other Soft Computing Techniques: Biomedical and Related Applications, Springer, Cham, Switzerland.

In our previous papers, we analyzed the idea of using light signals of three basic color -- red, green, and blue -- to speed up computations, in particular fuzzy-related computations. A natural question is: why red, green, and blue? Why not select some other colors: e.g., from the wavelength viewpoint, green is much closer to blue than to green, so why not select colors whose distribution is more even? In this paper, we show that if we consider this problem from the information viewpoint, then the corresponding equal-information criterion indeed implies that the intermediate wavelength should be closer to the smaller of the two remaining wavelengths than to the larger of these two. This result also explains why in human perception, green is closer to blue than to red. It also partially explains why green-blue color blindness it not the most frequent one -- which, from the viewpoint of differences in wavelength, sounds paradoxical.

To appear in: Nguyen Hoang Phuong, Nguyen Thi Huyen Chau, and Vladik Kreinovich (eds.), Explainable Artificial Intelligence and Other Soft Computing Techniques: Biomedical and Related Applications, Springer, Cham, Switzerland.

In engineering designs, we usually need to make sure that the values of some characteristics $y$ do not exceed a certain threshold y₀ -- e.g., that the stress at each location does not exceed a certain critical value. Usually, we know how each of these characteristics y depends on the design parameters x₁, ..., x_n, i.e., we know the function y=f(x₁,...,x_n). However, it is not enough to use the nominal values of the design parameters in our analysis, since the actual values are, in general, somewhat different from the nominal values. Often, the only information that we have about the actual values of the parameters x_i are tolerance intervals [x^&minus_i,x⁺_i]. It is therefore possible to compute the range [y^&minus,y^&minus] of the function f(x₁,...,x_n) on these intervals. The problem is that the resulting worst-case interval is too wide, it includes many value y that are not practically possible -- since it is highly improbable that all parameters attain their extreme values at the same time. Using this too-wide interval would lead to unnecessarily complicated and expensive design. It is therefore desirable to come up with a narrower interval. In this paper, we first provide a recommendation for computing such a narrower interval based on heuristic fuzzy-logic-based ideas. We also describe an alternative mathematically justified approach and show that it leads to exact same recommendations.

To appear in: Vladik Kreinovich, Woraphon Yamaka, and Supanika Leurcharusmee (eds.), Data Science for Econometrics and Related Topics, Springer, Cham, Switzerland.

In the 1950s, the future Nobelist Lloyd Shapley solved the problem of how to fairly divide the common gain. Namely, he showed that some reasonable requirements determine a unique division -- which is now known as the Shapley value. The main limitation of Shapley's solution is that it assumes that for each subgroup of the original group of participants, we know exactly how much this group could gain if it acted by itself, without involving others. In practice, we rarely know these exact values. At best, we know the bounds on each such value -- i.e., in other words, an interval that contains this value -- or even have no information about some of these values at all. In this paper, we show that a natural modification of Shapley's conditions enables us to extend Shapley's formulas to this realistic case, when we have interval uncertainty and partial information.

For each geographic location, its seismicity level is usually determined by how close this location is to the boundaries of tectonic plates. However, there is one notable exception: while Ireland and Britain are at approximately the same distance from such boundaries, the seismicity level in Ireland is much lower than in Britain. A recent paper provided a partial explanation for this phenomenon: namely, it turns out that the lithosphere under Ireland is unusually thick, and this can potentially lead to lower seismicity. However, the current explanation of the relation between the lithosphere thickness and seismicity level strongly depends on the specific details of the corresponding hypothetical mechanism. In this paper, we provide a general geometric explanation of this relation, an explanation that does not depend on the specific geophysical details.

Recent observations have shown that the angles between the Galaxy Center filaments and the Galactic plane follow a bimodal distribution: a large number of filaments are approximately orthogonal to the Galactic plane, a large number of filaments are approximately parallel to the Galactic plane, and much fewer filaments have other orientations. In this paper, we show this bimodal distribution can be explained by natural geometric symmetries.

To appear in: Nguyen Ngoc Thach, Nguyen Duc Trung, Doan Thanh Ha, and Vladik Kreinovich, Artificial Intelligence and Machine Learning for Econometrics: Applications and Regulation (and Related Topics), Springer, Cham, Switzerland.

Traditional decision theory recommendation about making a decision assume that we know both the probabilities of different outcomes of each possible decision, and we know the utility function -- that describes the decision maker's preferences. Sometimes, we can make a recommendation even when we only have partial information about utility. Such cases are known as cases of stochastic dominance. In other cases, in addition to not knowing the utility function, we also only have partial information about the probabilities of different outcomes. For example, we may only known bounds on the outcomes (case of interval uncertainty) or bounds on the values of the cumulative distribution function (case of p-box uncertainty). In this paper, we extend known stochastic dominance results to these two cases.

To appear in: Vladik Kreinovich, Woraphon Yamaka, and Supanika Leurcharusmee (eds.), Data Science for Econometrics and Related Topics, Springer, Cham, Switzerland.

In the State of Alaska there is no state income tax. Instead, there is a negative tex: every year every resident gets some money from the state. At present, every resident -- from the poorest to the richest -- gets the exact same amount of money: in 2024, it is expected to be around $1500. A natural question is: Is this fair? Maybe poor people should get more since their needs are greater? Maybe the rich people should get proportionally more, since fairness means equal added happiness for all, and for rich people, extra $1500 is barely noticeable? There have been many ethical discussions about this. In this paper, we analyze the problem from the mathematical viewpoint, and we show that the current arrangement, while not exactly the most fair, is close to the fair one -- at least much closer to fairness than the alternative proportional arrangement.

To appear in: Evgeny Dantsin and Vladik Kreinovich (eds.), Uncertainty Quantification and Uncertainty Propagation under Traditional and AI-Based Data Processing (and Related Topics): Legacy of Grigory Tseytin, Springer, Cham, Switzerland, to appear.

In many practical situations, for each of two classifications, we know the probabilities that a randomly selected object belong to different categories. For example, we know what proportion of people are below 20 years old, what proportion is between 20 and 30, etc., and we also know what proportion of people earns less than 10K, between 10K and 20K, etc. In such situations, we are often interested in proportion of people who are classified by two classifications into two given categories. For example, we are interested in the proportion of people whose age is between 20 and 30 and whose income is between 10K and 20K. If we do not have detailed records of all the objects, we select a small sample and count how many objects from this sample belong to each pair of categories. The resulting proportions are a good first-approximation estimate for the desired proportion. However, for a random sample proportions of each category are, in general, somewhat different from the proportions in the overall population. Thus, the first-approximation estimates need to be adjusted, so that they fit with the overall-population values. The problem of finding proper adjustments is known as the proportional fitting problem. There exist many efficient iterative algorithms for solving this problem, but it is still desirable to find classes for which even faster algorithms are possible. In this paper, we show that for the case when one of the classifications has only two categories, the proportional fitting problem can be reduced to solving a polynomial equation of order equal to number n of categories of the second classification. So, for n = 2, 3, 4, explicit formulas for solving quadratic, cubic, and quartic equations lead to explicit solutions for the proportional fitness problem. For n > 4, fast algorithms for solving polynomial equations lead to fast algorithms for solving the proportional fitness problem.

To appear in: Evgeny Dantsin and Vladik Kreinovich (eds.), Uncertainty Quantification and Uncertainty Propagation under Traditional and AI-Based Data Processing (and Related Topics): Legacy of Grigory Tseytin, Springer, Cham, Switzerland, to appear.

Many modern mathematical proofs are very complex, checking them is difficult; as a result, errors sneak into published proofs, even into proofs published in highly reputable journals. Sometimes, the errors are repairable, but sometimes, it turns out that the supposedly proven result is actually wrong. When the error is not noticed for some time, the published result is used to prove many other results -- and when the error is eventually found, all these new results are invalidated. This happened several times. Since it is not realistic to more thoroughly check all the proofs, and we want to minimize the risk of errors, it is desirable to come up with some methods to select the most suspicious proofs -- so that we can be more attentive when checking those. One such heuristic -- developed by mathematicians -- is that if subsequent results are too easy to obtain, the proof most probably has errors. This empirical heuristic works in many cases, which leads to a natural question: Why does it work? In this paper, we provide a possible explanation for this heuristic's success.

To appear in: Evgeny Dantsin and Vladik Kreinovich (eds.), Uncertainty Quantification and Uncertainty Propagation under Traditional and AI-Based Data Processing (and Related Topics): Legacy of Grigory Tseytin, Springer, Cham, Switzerland, to appear.

Many reasonable conditions have been formulated for a fuzzy "and"-operation: idempotency, commutativity, associativity, etc. It is known that the only "and"-operation that satisfies all these conditions is minimum, but minimum is not the most adequate description of expert's "and", and it often does not lead to the best control or the best decision. Many other more adequate "and"-operations (t-norms) have been proposed and effectively used, but they do not satisfy the natural idempotency condition. In this paper, we show that a small relaxation of the usual description of "and"-operations leads to the possibility of non-minimum idempotent operations. We also show that another natural condition -- of normalization invariance -- uniquely determines the resulting "and"-operation. This new "and"-operation is not only more intuitive, it leads to better application results.

To appear in: Evgeny Dantsin and Vladik Kreinovich (eds.), Uncertainty Quantification and Uncertainty Propagation under Traditional and AI-Based Data Processing (and Related Topics): Legacy of Grigory Tseytin, Springer, Cham, Switzerland, to appear.

It is known that quantum field theories that describe fields in our usual 4-dimensional space-times are not fully consistent: they predict meaningless infinite values for some physical quantities. There are some known tricks to avoid such infinities, but it is definitely desirable to have a fully consistent theory, a theory that would produce correct results without having to use additional tricks. It turns out that the only way to have such a theory is to consider space-times of higher dimensions, the smallest of which is 10. There are complex mathematical reasons for why 10 is the smallest such dimension. However, from the physics viewpoint, it would be nice to supplement these complex mathematical reasons with a more commonsense explanation. There exists a particle-based commonsense explanation for this phenomenon. In this paper, we supplement it with a field-based commonsense explanation.

To appear in: Evgeny Dantsin and Vladik Kreinovich (eds.), Uncertainty Quantification and Uncertainty Propagation under Traditional and AI-Based Data Processing (and Related Topics): Legacy of Grigory Tseytin, Springer, Cham, Switzerland, to appear.

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.

To appear in Proceedings of the 10th International Workshop on Reliable Engineering Computing REC'2024, Beijing, China, October 26-27, 2024, pp. 5-12.

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. As a result, the computation time for uncertainty propagation is several times larger than data processing itself. This is a very critical issue for data processing algorithms that take a lot of computational steps -- such as modern deep learning-based AI techniques, for which a several-times increase in computation time is not feasible. At first glance, the situation may seem hopeless. Good news is that there is another problem with modern AI algorithms: usually, once they learn, their weights are frozen, and they stop learning -- as a result, the quality of their answers decreases with time. This is good news because, as we show, solving the second problem -- by allowing at least one learning step for each new use of the model -- helps to also come up with an efficient uncertainty propagation algorithm.

To appear in: Evgeny Dantsin and Vladik Kreinovich (eds.), Uncertainty Quantification and Uncertainty Propagation under Traditional and AI-Based Data Processing (and Related Topics): Legacy of Grigory Tseytin, Springer, Cham, Switzerland, to appear.

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 where we can find a natural analogy with colors: fuzzy control and quantum computing. Since the analogy with fuzzy control has already led to efficient applications, we hope that the newly discovered analogy with quantum computing will also lead to computational speedup.

To appear in: Evgeny Dantsin and Vladik Kreinovich (eds.), Uncertainty Quantification and Uncertainty Propagation under Traditional and AI-Based Data Processing (and Related Topics): Legacy of Grigory Tseytin, Springer, Cham, Switzerland, to appear.

In many practical situations, it is desirable to select the control parameters x₁, ..., x_n in such a way that the resulting quantities y₁, ..., y_m of the system lie within desired ranges. In such situations, we usually know the general formulas describing the dependence of y_i on x_j, 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 y_i'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 inclusions with probability 1, a natural idea is to select parameters for which the probability that all inclusions are satisfied is the largest. To implement this idea, we need to select a probability distribution on the set of all tuples. Since we have no reason to believe that some tuples are more probable than others, it is reasonable to assume that all tuples are equally probable, i.e., that we have a uniform distribution on the set of all tuples. Interestingly, this idea leads to the same recommendation as was proposed -- based on heuristic fuzzy-logic-based arguments -- in a recent paper by Piegat. An important remaining open problem is how to efficiently compute the recommended solution.

To appear in Proceedings of the 10th International Workshop on Reliable Engineering Computing REC'2024, Beijing, China, October 26-27, 2024, pp. 13-25.

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 two related approaches can lead to such a foundation: the approach based on computational complexity and the symmetry-based approach. As a result, we get an explanation of why, in general, fuzzy and neural techniques are so successful, and why specific version of these techniques are empirically the most successful, such as ReLU activation function and the use of piecewise linear membership functions in fuzzy approach.

To appear in Proceedings of the 10th International Workshop on Reliable Engineering Computing REC'2024, Beijing, China, October 26-27, 2024, pp. 89-99.

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 of possible values of the following states. It turns out that we can feasible predict the future state one moment ahead, but predicting two moments ahead is already NP-hard -- meaning that (unless P = NP), no feasible algorithm can preform these predictions for all possible linear dynamical systems under interval uncertainty.

To appear in: Evgeny Dantsin and Vladik Kreinovich (eds.), Uncertainty Quantification and Uncertainty Propagation under Traditional and AI-Based Data Processing (and Related Topics): Legacy of Grigory Tseytin, Springer, Cham, Switzerland, to appear.

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, we provide a possible theoretical explanation for this two-sigma phenomenon. In our explanation, we use another previously-difficult-to-explain empirical fact: that the grade distribution is often bimodal -- and we explain this auxiliary fact too. In view of the above, we hope that our explanations will eventually lead to a more effectively use of emerging technologies in education.

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

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) success.

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 English, and 0.17 for Spanish. We also provide examples of additional possible pragmatic functions of reduction in English, and various discussion, observations and speculations.

To appear in: Vladik Kreinovich, Woraphon Yamaka, and Supanika Leurcharusmee (eds.), Data Science for Econometrics and Related Topics, Springer, Cham, Switzerland.

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 finding the smallest number of constraints.

To appear in Proceedings of the NAFIPS International Conference on Fuzzy Systems, Soft Computing, and Explainable AI NAFIPS'2024, South Padre Island, Texas, May 27-29, 2024

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.

To appear in: Proceedings of the XXIV World Congress of the International Measurement Confederation IMEKO, Hamburg, Germany, August 26-29, 2024.

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 actual 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 nearby locations. It is known that taking such relation into account can lead to more accurate estimates for physical quantities. In this paper, we describe a general methodology for taking these relations into account. We also show how this methodology can help to detect faulty measuring instruments -- thus increasing the reliability of the measurement results.

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

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.

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

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.

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

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 information about the length of each number. In view of this, the first natural question is whether a varying-length representation can lead to a drastic decrease in needed computer space. Another natural question is related to the fact that while potentially, allowing number of bits which is not proportional to 8 bits per byte will save even more space, this would require a drastic change in computer architecture, since the current architecture is based on bytes. So will going from bytes to bits be worth it -- will it save much space? In this paper, we provide answers to both questions.

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

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.

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

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.

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

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 long as the corresponding comparison between two behaviors does not change under rotations, shifts, and permutations.

To appear in Proceedings of the NAFIPS International Conference on Fuzzy Systems, Soft Computing, and Explainable AI NAFIPS'2024, South Padre Island, Texas, May 27-29, 2024

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.

To appear in Proceedings of the NAFIPS International Conference on Fuzzy Systems, Soft Computing, and Explainable AI NAFIPS'2024, South Padre Island, Texas, May 27-29, 2024

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 not provide us with a good understanding of how unequal are incomes of the majority of folks. For this purpose, Theil introduced decomposable modifications of the standard deviation that is now called Theil indices. Crudely speaking, these indices are based on using logarithm instead of the square. Other researchers found other another decomposable modifications that use power law. In this paper, we provide a complete description of all decomposable versions of the Theil index. Specifically, we prove that the currently known functions are the only one for which the corresponding versions of the Theil index are decomposable -- so no other decomposable versions are possible. A similar result was previously proven under the additional assumption of linearity; our proof shows that this result is also true in the general case, without assuming linearity.

To appear in Proceedings of the NAFIPS International Conference on Fuzzy Systems, Soft Computing, and Explainable AI NAFIPS'2024, South Padre Island, Texas, May 27-29, 2024

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 such knowledge into precise terms, Zadeh pioneered a technique that he called fuzzy. Fuzzy techniques have many successful applications; however, expert statements are subjective; in contrast to measurement results, they do not come with guaranteed accuracy. In this paper, we show that by using fuzzy techniques, we can translate imprecise expert knowledge into precise probabilistic terms -- thus allowing to combine this knowledge with measurement results into a model with guaranteed accuracy.

To appear in Proceedings of the NAFIPS International Conference on Fuzzy Systems, Soft Computing, and Explainable AI NAFIPS'2024, South Padre Island, Texas, May 27-29, 2024

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.

To appear in Proceedings of the NAFIPS International Conference on Fuzzy Systems, Soft Computing, and Explainable AI NAFIPS'2024, South Padre Island, Texas, May 27-29, 2024

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. At best, we know the intervals of possible values of these quantities. For different values from these intervals, we get, in general, different probabilities. It is desirable to find the range of all possible values of these probabilities. In this paper, we provide a feasible algorithm for computing these ranges.

To appear in: Martine Ceberio and Vladik Kreinovich (eds.), "Uncertainty, Constraints, AI, and Decision Making", Springer, Cham, Switzerland.

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.

To appear in Proceedings of the NAFIPS International Conference on Fuzzy Systems, Soft Computing, and Explainable AI NAFIPS'2024, South Padre Island, Texas, May 27-29, 2024

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.

To appear in: Martine Ceberio and Vladik Kreinovich (eds.), "Uncertainty, Constraints, AI, and Decision Making", Springer, Cham, Switzerland.

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.

To appear in: Marie-Jeanne Lesot, Marek Reformat, Susana Vieira, Joao P. Carvalho, Fernando Batista, Bernadette Bouchon-Meunier, and Ronald R. Yager (eds.), Information Processing and Management of Uncertainty in Knowledge-Based Systems, Proceedings of the 20th International Conference IPMU 2024, Lisbon, Portugal, July 22-26, 2024.

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 plan the measurements and to quantify and propagate the resulting uncertainty and reliability.

To appear in: Yuriy P. Kondratenko (ed.), Research Tendencies and Prospect Domains for AI Development and Implementation, River Publishers, Denmark.

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, we have a straight path of speeding up computations: by learning how to use molecules, atoms, and then elementary particles as building blocks of a computational device. But what if we reach the size of an elementary particle? At first glance, it may seem that we will then reach an absolute limit of how fast a computer can be. However, as we show in this paper, we can potentially speed up computations even further -- by using the internal structure of elementary particles: e.g., the fact that protons and neutrons consist of quarks. Interestingly, the corresponding mathematics is very similar to what is called color optical computing -- the use of light of different colors in computations.

Published in: Proceedings of the IEEE 18th International Symposium on Applied Computational Intelligence and Informatics SACI 2024, Siofok, Hungary, and Timisoara, Romania, May 21-25, 2024, pp. 127-132.

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 this paper, we show that fuzzy ideas lead to the selection of logarithmic re-scaling, which indeed works very well in video analysis -- and which also explains Fechner Law in psychology, that our perception of difference between two stimuli is determined by the difference between the logarithms of their intensities.

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 for which there is no convincing justification. In this paper, we show that while we cannot directly apply the maximum entropy approach to the skewness situation, we can apply it approximately -- with any approximation accuracy we want -- and get a meaningful answer to the above selection problem.

To appear in: Martine Ceberio and Vladik Kreinovich (eds.), "Uncertainty, Constraints, AI, and Decision Making", Springer, Cham, Switzerland.

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 extension is indeed possible.