Vladik Kreinovich

• PhD
• Professor (Full) at The University of Texas at El Paso

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 signific...

For normalized fuzzy sets, intersection is, in general, not normalized. So, if we want to limit ourselves to normalized fuzzy sets, we need to normalize the intersection. It is known that for algebraic product, the normalized intersection is associative, and that for many other “and”-operations (t-norms), normalized intersection is not associative....

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...

This volume emphasizes techniques of optimal transport statistics, but it also describes and uses other econometric techniques, ranging from more traditional statistical techniques to more innovative ones such as quantiles (in particular, multidimensional quantiles), maximum entropy approach, and machine learning. Applications range from general an...

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...

Purpose When the probability of each model is known, a natural idea is to select the most probable model. However, in many practical situations, the exact values of these probabilities are not known; only the intervals that contain these values are known. In such situations, a natural idea is to select some probabilities from these intervals and to...

In the ideal world, the number of seats that each region or each community gets in a representative body should be exactly proportional to the population of this region or community. However, since the number of seats allocated to each region or community is whole, we cannot maintain the exact proportionality. Not only this leads to a somewhat unfa...

In many practical situations, we need to measure the value of a cumulative quantity, i.e., a quantity that is obtained by adding measurement results corresponding to different spatial locations. How can we select the measuring instruments so that the resulting cumulative quantity can be determined with known accuracy–and, to avoid unnecessary expen...

Recently, astronomers discovered spiral arms around a star. While their shape is similar to the shape of the spiral arms in the galaxies, however, because of the different scale, galaxy-related physical explanations of galactic spirals cannot be directly applied to explaining star-size spiral arms. In this paper, we show that, in contrast to more s...

There is a reasonably accurate empirical formula that predicts, for two words i and j, the number \(X_{ij}\) of times when the word i will appear in the vicinity of the word j. The parameters of this formula are determined by using the weighted least square approach. Empirically, the predictions are the most accurate if we use the weights proportio...

It is known that self-esteem helps solve problems. From the algorithmic viewpoint, this seems like a mystery: a boost in self-esteem does not provide us with new algorithms, does not provide us with ability to compute faster—but somehow, with the same algorithmic tools and the same ability to perform the corresponding computations, students become...

In many practical situations—e.g., when preparing examples for a machine learning algorithm—we need to label a large number of images or speech recordings. One way to do it is to pay people around the world to perform this labeling; this is known as crowdsourcing. In many cases, crowd-workers generate not only answers, but also their degrees of con...

Many immunodepressive drugs have an unusual side effect on the patient’s mood: they often make the patient happier. This side effect has been observed for many different immunodepressive drugs, with different chemical composition. Thus, it is natural to conclude that there must be some general reason for this empirical phenomenon, a reason not rela...

In many practical situations when we need to select the best submission—the best paper, the best candidate, etc.—there are so few experts that we cannot simply dismiss all the experts who have conflict of interest: we do not want them to judge their own submissions, but we would like to take into account their opinions of all other submissions. How...

To analyze the effect of pollution on marine life, it is important to know how exactly the concentration of toxic substances decreases with time. There are several semi-empirical formulas that describe this decrease. In this paper, we provide a theoretical explanation for these empirical formulas.

In a computer, all the information about an object is described by a sequence of 0s and 1s. At any given moment of time, we only have partial information, but as we perform more measurements and observations, we get longer and longer sequence that provides a more and more accurate description of the object. In the limit, we get a perfect descriptio...

Starting with the 1980s, a popular US satirical radio show described a fictitious town Lake Wobegon where “all children are above average”—parodying the way parents like to talk about their children. This everyone-above-average situation was part of the fiction since, if we interpret the average in the precise mathematical sense, as average over al...

Most practical problems lead either to solving a system of equation or to optimization. From the computational viewpoint, both classes of problems can be reduced to each other: optimization can be reduced to finding points at which all partial derivatives are zeros, and solving systems of equations can be reduced to minimizing sums of squares. It i...

One of the most prospective aerospace engines is a Dielectric Barrier Discharge (DBD) thruster–an effective electric engine without moving parts. The efficiency of this engine depends on the proper selection of the corresponding electric field. To make this selection, we need to know, in particular, how its thrust depends on the atmospheric pressur...

For multi-tasking optimization problems, it has been empirically shown that the most effective resource allocation is attained when we assume that the gain of each task logarithmically depends on the computation time allocated to this task. In this paper, we provide a theoretical explanation for this empirical fact.

In probability theory, rare events are usually described as events with low probability p, i.e., events for which in N observations, the event happens \(n(N)\sim p\cdot N\) times. Physicists and philosophers suggested that there may be events which are even rarer, in which n(N) grows slower than N. However, this idea has not been developed, since i...

Predatory birds play an important role in an ecosystem. It is therefore important to study their hunting behavior, in particular, the distribution of their waiting time. A recent empirical study showed that the waiting time is distributed according to the power law. In this paper, we use natural invariance ideas to come up with a theoretical explan...

One of the main motivations for using artificial neural networks was to speed up computations. From this viewpoint, the ideal configuration is when we have a single nonlinear layer: this configuration is computationally the fastest, and it already has the desired universal approximation property. However, the last decades have shown that for many p...

At first glance, from the general decision-theory viewpoint, hate (and other negative feelings towards each other) makes no sense, since they decrease the utility (i.e., crudely speaking, level of happiness) of the person who experiences these feelings. Our detailed analysis shows that there are situations when such negative feelings make perfect s...

Business gurus recommend that an organization should have, in addition to clearly described realistic goals, also additional aspirational goals—goals for which we may not have resources and which most probably will not be reached at all. At first glance, adding such a vague goal cannot lead to a drastic change in how the company operates, but surpr...

Strangely enough, investors invest in high-risk low-profit enterprises as well. At first glance, this seems to contradict common sense and financial basics. However, we show that such investments make perfect sense as long as the related risks are independent from the risks of other investments. Moreover, we show that an optimal investment portfoli...

In many practical situation, control experts can only formulate their experience by using imprecise (“fuzzy”) words from natural language. To incorporate this knowledge in automatic controllers, Lotfi Zadeh came up with a methodology that translates the informal expert statements into a precise control strategy. This methodology – and its following...

It has been recently shown that in some applications, e.g., in ship navigation near a harbor, it is convenient to use combinations of basic colors – red, green, and blue – to represent different fuzzy degrees. In this paper, we provide a natural explanation for the efficiency of this empirical fact: namely, we show: (1) that it is reasonable to con...

This work is devoted to the creation of optical logic gates for the synthesis of efficient systems with a high processing speed of fuzzy data, as well as a wide possibility of parallel computing. Logical operations are based on the use of a light emitter of a certain color as a fuzzy variable - a carrier of logical information and the basis for con...

This paper is dedicated to the development of intelligent techniques of optical computing for real-time decision support systems (DSS) with a large array of fuzzy input data. Currently offered as an alternative to the binary systems, fuzzy optical computing devices are very complex. To increase the efficiency of logical systems in the formation and...

Traditional analysis of uncertainty of the result of data processing assumes that all measurement errors are independent. In reality, there may be common factor affecting these errors, so these errors may be dependent. In such cases, the independence assumption may lead to underestimation of uncertainty. In such cases, a guaranteed way to be on the...

Based on the rotation of the stars around a galaxy center, one can estimate the corresponding gravitational acceleration—which turns out to be much larger than what Newton’s theory predicts based on the masses of all visible objects. The majority of physicists believe that this discrepancy indicates the presence of “dark” matter, but this idea has...

In this paper, we show that requirements that computations be fast and noise-resistant naturally lead to what we call color-based optical computing.

When we know for sure which values are possible and which are not, we have crisp uncertainty—of which interval uncertainty is a usual case. In practice, we are often not 100% sure about our knowledge, i.e., we have fuzzy uncertainty—i.e., we have fuzzy knowledge, of which crisp is a particular case. Usually, general problems are more difficult to s...

Many people remember Lofti Zadeh’s mantra—that everything is a matter of degree. This was one of the main principles behind fuzzy logic. What is somewhat less remembered is that Zadeh also used another important principle—that there is a need for simplicity. In this paper, we show that together, these two principles can generate the main ideas behi...

One of the main objectives of fuzzy control is to translate expert rules –formulated in imprecise (“fuzzy”) words from natural language– into a precise control strategy. This translation is usually done in two steps. First, we apply a fuzzy control methodology to get a rough approximation of the expert’s control strategy, and then we tune the resul...

Mục đích – Năm 1951, Kenneth Arrow đã chứng minh rằng không thể có một quy trình ra quyết định theo nhóm đáp ứng các yêu cầu hợp lý như tính công bằng. Về mặt lý thuyết, đây là một kết quả tuyệt vời – rất xứng đáng với giải thưởng Nobel đã được trao cho Giáo sư Arrow. Tuy nhiên, từ quan điểm thực tế, câu hỏi vẫn còn đó – vậy chúng ta nên đưa ra quy...

In general, the more measurements we perform, the more information we gain about the system and thus, the more adequate decisions we will be able to make. However, in situations when we perform measurements to check for safety, the situation is sometimes opposite: the more additional measurements we perform beyond what is required, the worse the de...

In high performance computing, when we process a large amount of data, we do not have much information about the dependence between measurement errors corresponding to different inputs. To gauge the uncertainty of the result of data processing, the two usual approaches are: the interval approach, when we consider the worst-case scenario, and the pr...

In many practical situations, we need to determine the dependence between different quantities based on the empirical data. Several methods exist for solving this problem, including neural techniques and different versions of fuzzy techniques: type-1, type-2, etc. In some cases, some of these techniques work better, in other cases, other methods wo...

Economic and financial phenomena are highly complex and non-linear. However, surprisingly, in many cases, these phenomena are accurately described by linear models-or, sometimes, by piecewise linear ones. In this paper, we show that fuzzy techniques can explain the unexpected efficiency of linear and piecewise linear models: namely, we show that a...

The fact that \(\infty \) is actively used as a symbol for infinity shows that this symbol is probably reasonable in this role, but why? In this paper, we provide a possible explanation for why this is indeed a reasonable symbol for infinity.

To elicit people’s opinions, we usually ask them to mark their degree of satisfaction on a scale—e.g., from 0 to 5 or from 0 to 10. Often, people are unsure about the exact degree: 7 or 8? To cover such situations, it is desirable to elicit not a single value but an interval of possible values. However, it turns out that most people are not comfort...

Why are we using the decimal system to describe numbers? Why all over the world, communities with more than 150 folks tend to split? In this paper, we show that both phenomena—as well as some other phenomena—can be explained if we take into account the seven plus minus two law, according to which a person can keep in immediate memory from 5 to 9 it...

In most practical applications, we approximate the spatial dependence by smooth functions. The main exception is geosciences, where, to describe, e.g., how the density depends on depth and/or on spatial location, geophysicists divide the area into regions on each of which the corresponding quantity is approximately constant. In this paper, we provi...

The paper describes and explains the teaching strategy of Iosif Yakovlevich Verebeichik, a successful mathematics teacher at special mathematical high schools—schools for students interested in and skilled in mathematics. The resulting strategy seems counterintuitive and contrary to all the pedagogical advice. Our explanation is not complete: it wo...

To simplify the design of compilers, Noam Chomsky proposed to first transform a description of a programming language—which is usually given in the form of a context-free grammar—into a simplified “normal” form. A natural question is: why this specific normal form? In this paper, we provide an answer to this question.

In many practical situations, we have a large number of objects, too many to be able to thoroughly analyze each of them. To get a general understanding, we need to select a representative sample. For us, this problem was motivated by the need to analyze the possible effect of an earthquake on building in El Paso, Texas. In this paper, we provide a...

The usual formulas for gauging the quality of a classification method assume that we know the ground truth, i.e., that for several objects, we know for sure to which class they belong. In practice, we often only know this with some degree of certainty. In this paper, we explain how to take this uncertainty into account when gauging the quality of a...

According to the general idea of quantization, all physical dependencies are only approximately deterministic, and all physical “constants” are actually varying. A natural conclusion—that some physicists made—is that Planck’s constant (that determines the magnitude of quantum effects) can also vary. In this paper, we use another general physics ide...

Reasonably recent experiments show that unhappiness is strongly correlated with the excessive interaction between two parts of the brain—amygdala and hippocampus. At first glance, in situations when outside signals are positive, additional interaction between two parts of the brain that get signals from different sensors should only reinforce the p...

Why do people become addicted, e.g., to gambling? Experiments have shown that simple lotteries, in which we can win a small prize with a certain probability, and not addictive. However, if we add a second possibility—of having a large prize with a small probability—the lottery becomes highly addictive to many participants. In this paper, we provide...

Historically, to describe numbers, some cultures used bases much larger than our usual base 10, namely, bases 20, 40, and 60. There are explanations for base 60, there is some explanation for base 20, but base 40—used in medieval Russia—remains largely a mystery. In this paper, we provide a possible explanation for all these three bases, an explana...

Since in the physical world, most dependencies are smooth (differentiable), traditionally, smooth functions were used to approximate these dependencies. In particular, neural networks used smooth activation functions such as the sigmoid function. However, the successes of deep learning showed that in many cases, non-smooth activation functions like...

While we currently only observe 3 spatial dimensions, according to modern physics, our space is actually at least 10-dimensional. In this paper, on different versions of the multi-D spatial models, we analyze how the existence of the additional spatial dimensions can help computations. It turns out that in all the versions, there is some speed up—m...

The main idea behind semi-supervised learning is that when we do not have enough human-generated labels, we train a machine learning system based on what we have, and we add the resulting labels (called pseudo-labels) to the training sample. Interesting, this idea works well, but why is somewhat a mystery: we did not add any new information so why...

The main idea behind a smart grid is to equip the grid with a dense lattice of sensors monitoring the state of the grid. If there is a fault, the sensors closer to the fault will detect larger deviations from the normal readings that sensors that are farther away. In this paper, we show that this fact can be used to locate the fault with high accur...

In situations when we have a perfect knowledge about the outcomes of several situations, a natural idea is to select the best of these situations. For example, among different investments, we should select the one with the largest gain. In practice, however, we rarely know the exact consequences of each action. In some cases, we know the lower and...

AlphaZero and its extension MuZero are computer programs that use machine-learning techniques to play at a superhuman level in chess, go, and a few other games. They achieved this level of play solely with reinforcement learning from self-play, without any domain knowledge except the game rules. It is a natural idea to adapt the methods and techniq...

What is 1/0: Students are first taught—in elementary school—that it is undefined, then—in calculus—then it is infinity. In both cases, the answer is usually provided based on abstract reasoning. But what about the practical meaning? In this paper, we show that, depending on the specific practical problem, we can have different answers to this quest...

In his unpublished paper, the famous logician Kurt Gödel provided arguments in favor of the existence of God. These arguments are presented in a very formal way, which makes them difficult to understand to many interested readers. In this paper, we describe a simplifying modification of Gödel’s proof which will hopefully make it easier to understan...

According to the analysis by the French philosopher Jean-Paul Sartre, the famous French poet and essayist Charles Baudelaire described (and followed) two main—somewhat unusual—ideas about art: that art should be vague, and that to create an object of art, one needs to aim for uniqueness. In this paper, we provide an algorithm-based explanation for...

Nobelist physicist Lev Landau was known for applying mathematical and physical reasoning to human relations. His advices may have been somewhat controversial, but they were usually well motivated. However, there was one advice for which no explanation remains—that a person should not marry his/her first and second true loves, and only start thinkin...

It is a known empirical fact that people overestimate small probabilities. This fact seems to be inconsistent with the fact that we humans are the product of billions years of improving evolution—and that we therefore perceive the world as accurately as possible. In this paper, we provide a possible explanation for this seeming contradiction.

In this paper, we show that many seemingly irrational Biblical ideas can actually be rationally interpreted: that God is everywhere, that we can only say what God is not, that God’s name is holy, why cannot you bless as many people as you want, etc. We do not insist on our interpretations, there probably are many others, our sole objective was to s...

Sigmund Freud famously placed what he called Oedipus complex at the center of his explanation of psychological and psychiatric problems. Freund’s analysis was based on anecdotal evidence and intuition, not on solid experiments—as a result, for a long time, many psychologists dismissed the universality of Freud’s findings. However, lately, experimen...

In many practical situations, we need to estimate our degree of belief in a statement \( A\, \& \,B\) when the only thing we know are the degrees of belief a and b in combined statements A and B. An algorithm for this estimation is known as an “and”-operation, or, for historical reasons, a t-norm. Usually, “and”-operations are selected in such a wa...

Among the most efficient characteristics of a probability distribution are its moments and, more generally, generalized moments. One of the most adequate numerical characteristics describing human behavior is expected utility. In both cases, the corresponding characteristic is the sum of results of applying appropriate nonlinear functions applied t...

Please send your abstracts (or copies of papers that you want to see reviewed here) to vladik@utep.edu, or by regular mail to Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA…

Somewhat surprisingly, several formulas of quantum physics—the physics of micro-world—provide a good first approximation to many social phenomena, in particular, to many economic phenomena, phenomena which are very far from micro-physics. In this paper, we provide three possible explanations for this surprising fact. First, we show that several for...

At present, the most successful machine learning technique is deep learning, that uses rectified linear activation function (ReLU) \(s(x) = \max (x,0)\) as a non-linear data processing unit. While this selection was guided by general ideas (which were often imprecise), the selection itself was still largely empirical. This leads to a natural questi...

Predictions are usually based on what is called laws of nature: many times, we observe the same relation between the states at different moments of time, and we conclude that the same relation will occur in the future. The more times the relation repeats, the more confident we are that the same phenomenon will be repeated again. This is how Newton’...

At present, the most efficient deep learning technique is the use of deep neural networks. However, recent empirical results show that in some situations, it is even more efficient to use “localized” learning—i.e., to divide the domain of inputs into sub-domains, learn the desired dependence separately on each sub-domain, and then “smooth” the resu...

We consider a general model of financial flows and prices with multiple sectors and instruments. Each sector optimizes the composition of assets and liabilities in its portfolio, whose utility is given by a quadratic function constrained to satisfy the accounting identity that appears in flow-of-funds accounts and the equilibrium conditions that gu...

This work is devoted to the creation of effective optical logic systems based on the use of light emitter of a certain color directly as a fuzzy variable — the carrier of logical information and the basis for building logical solutions by transforming light emitter with appropriate light filters. Optical processing of color information, which refle...

What is explainable AI? Why do we need explainable AI?

In the previous chapter, we showed how, for each possible value u of control, we can generate the degree μ(u) to which this value is reasonable. As a result, we get what we called a membership function (or a fuzzy set) μ(u).

What is machine learning: reminder. As we recall, machine learning means that