Physical Sciences and Mathematics Commons^™

What If Not All Interval-Valued Fuzzy Degrees Are Possible?, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the applications of intervals is in describing experts' degrees of certainty in their statements. In this application, not all intervals are realistically possible. To describe all realistically possible degrees, we end up with a mathematical question of describing all topologically closed classes of intervals which are closed under the appropriate minimum and maximum operations. In this paper, we provide a full description of all such classes.

Healthy Lifestyle Decreases The Risk Of Alzheimer Disease: A Possible Partial Explanation Of An Empirical Dependence, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

A recent paper showed that for people who follow all five healthy lifestyle recommendations, the risk of Alzheimer disease is only 40% of the risk for those who do not follow any of these recommendations, and that for people two or three of these recommendations, the risk is 63% of the not-followers risk. In this paper, we show that a relation between the two numbers -- namely, that 0.40 is the square of 0.63 -- can be naturally explained by a simple model.

When Can We Be Sure That Measurement Results Are Consistent: 1-D Interval Case And Beyond, Hani Dbouk, Steffen Schön, Ingo Neumann, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, measurements are characterized by interval uncertainty -- namely, based on each measurement result, the only information that we have about the actual value of the measured quantity is that this value belongs to some interval. If several such intervals -- corresponding to measuring the same quantity -- have an empty intersection, this means that at least one of the corresponding measurement results is an outlier, caused by a malfunction of the measuring instrument. From the purely mathematical viewpoint, if the intersection is non-empty, there is no reason to be suspicious, but from the practical viewpoint, if …

Common-Sense-Based Theoretical Explanation For An Empirical Formula Estimating Road Quality, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

The quality of a road is usually gauged by a group of trained raters; the resulting numerical value is known as the Present Serviceability Index (PSI). There are, however, two problems with this approach. First, while it is practical to use trained raters to gauge the quality of major highways, there are also numerous not-so-major roads, and there is not enough trained raters to gauge the quality of all of them. Second, even for skilled raters, their estimates are somewhat subjective: different groups of raters may estimate the quality of the same road segment somewhat differently. Because of these two …

Approximate Version Of Interval Computation Is Still Np-Hard, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

It is known that, in general, the problem of computing the range of a given polynomial on given intervals is NP-hard. For some NP-hard optimization problems, the approximate version -- e.g., if we want to find the value differing from the maximum by no more than a factor of 2 -- becomes feasible. Thus, a natural question is: what if instead of computing the exact range, we want to compute the enclosure which is, e.g., no more than twice wider than the actual range? In this paper, we show that this approximate version is still NP-hard, whether we want it …

Which Classes Of Bi-Intervals Are Closed Under Addition?, Olga Kosheleva, Vladik Kreinovich, Jonatan Contreras

Departmental Technical Reports (CS)

In many practical situations, uncertainty with which we know each quantity is described by an interval. In processing such data, it is useful to know that the sum of two intervals is always an interval. In some cases, however, the set of all possible value of a quantity is described by a bi-interval -- i.e., by a union of two intervals. It is known that the sum of two bi-intervals is not always a bi-interval. In this paper, we describe all the class of bi-intervals which are closed under addition -- i.e., for which the sum of bi-intervals is a …

Preference For Boys Does Not Necessarily Lead To A Gender Disbalance: A Realistic Example, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Intuitively, it seems that cultural preference for boys should lead to a gender disbalance -- more boys than girls. This disbalance is indeed what is often observed, and this disbalance is what many models predict. However, in this paper, we show, on a realistic example, that preference for boys does not necessarily lead to a gender disbalance: in our simplified example, boys are clearly preferred, but still there are exactly as many girls as there are boys.

Reward For Good Performance Works Better Than Punishment For Mistakes: Economic Explanation, Olga Kosheleva, Julio Urenda, Vladik Kreinovich

Departmental Technical Reports (CS)

How should we stimulate people to make them perform better? How should we stimulate students to make them study better? Many experiments have shown that reward for good performance works better than punishment for mistakes. In this paper, we provide a possible theoretical explanation for this empirical fact.

Why Some Powers Laws Are Possible And Some Are Not, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich, Olga Kosheleva, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

Many dependencies between quantities are described by power laws, in which y is proportional to x raised to some power a. In some application areas, in different situations, we observe all possible pairs (A,a) of the coefficient of proportionality A and of the exponent a. In other application areas, however, not all combinations (A,a) are possible: once we fix the coefficient A, it uniquely determines the exponent a. In such case, the dependence of a on A is usually described by an empirical logarithmic formula. In this paper, we show that natural scale-invariance ideas lead to a theoretical explanation for …

Formal Concept Analysis Techniques Can Help In Intelligent Control, Deep Learning, Etc., Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we show that formal concept analysis is a particular case of a more general problem that includes deriving rules for intelligent control, finding appropriate properties for deep learning algorithms, etc. Because of this, we believe that formal concept analysis techniques can be (and need to be) extended to these application areas as well. To show that such an extension is possible, we explain how these techniques can be applied to intelligent control.

How Expert Knowledge Can Help Measurements: Three Case Studies, Vladik Kreinovich

Departmental Technical Reports (CS)

In addition to measurement results, we often have expert estimates. These estimates provides an additional information about the corresponding quantities. However, it is not clear how to incorporate these estimates into a metrological analysis: metrological analysis is usually based on justified statistical estimates, but expert estimates are usually not similarly justified. One way to solve this problem is to calibrate an expert the same way we calibrate measuring instruments. In the first two case studies, we show that such a calibration indeed leads to useful result. The third case study provides an example of another use of expert knowledge in …

Neural Networks, Vladik Kreinovich

Departmental Technical Reports (CS)

A neural network is a general term for machine learning tools that emulate how neurons work in our brains.

Ideally, these tools do what we scientists are supposed to do: we feed them examples of the observed system's behavior, and hopefully, based on these examples, the tool will predict the future behavior of similar systems. Sometimes they do predict -- but in many other cases, the situation is not so simple.

The goal of this entry is to explain what these tools can and cannot do -- without going into too many technical details.

Absence Of Remotely Triggered Large Earthquakes: A Geometric Explanation, Laxman Bokati, Aaron A. Velasco, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that seismic waves from a large earthquake can trigger earthquakes in distant locations. Some of the triggered earthquakes are strong themselves. Interestingly, strong triggered earthquakes only happen within a reasonably small distance (less than 1000 km) from the original earthquake. Even catastrophic earthquakes do not trigger any strong earthquakes beyond this distance. In this paper, we provide a possible geometric explanation for this phenomenon.

Economics Of Reciprocity And Temptation, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich, Nguyen Ngoc Thach

Departmental Technical Reports (CS)

Behavioral economics has shown that in many situations, people's behavior differs from what is predicted by simple traditional utility-maximization economic models. It is therefore desirable to be able to accurately describe people's actual behavior. In some cases, the difference from the traditional models is caused by bounded rationality -- our limited ability to process information and to come up with a truly optimal solutions. In such cases, predicting people's behavior is difficult. In other cases, however, people actually optimize -- but the actual expression for utility is more complicated than in the traditional models. In such case, it is, in …

How The Proportion Of People Who Agree To Perform A Task Depends On The Stimulus: A Theoretical Explanation Of The Empirical Formula, Laxman Bokati, Vladik Kreinovich, Doan Thanh Ha

Departmental Technical Reports (CS)

For each task, the larger the stimulus, the larger proportion of people agree to perform this task. In many economic situations, it is important to know how much stimulus we need to offer so that a sufficient proportion of the people will agree to perform the needed task. There is an empirical formula describing how this proportion increases as we increase the amount of stimulus. However, this empirical formula lacks a convincing theoretical explanation, as a result of which practitioners are somewhat reluctant to use it. In this paper, we provide a theoretical explanation for this empirical formula, thus making …

Towards Fast And Understandable Computations: Which "And"- And "Or"-Operations Can Be Represented By The Fastest (I.E., 1-Layer) Neural Networks? Which Activations Functions Allow Such Representations?, Kevin Alvarez, Julio Urenda, Orsoly Csiszár, Gábor Csiszár, József Dombi, György Eigner, Vladik Kreinovich

Departmental Technical Reports (CS)

We want computations to be fast, and we want them to be understandable. As we show, the need for computations to be fast naturally leads to neural networks, with 1-layer networks being the fastest, and the need to be understandable naturally leads to fuzzy logic and to the corresponding "and"- and "or"-operations. Since we want our computations to be both fast and understandable, a natural question is: which "and"- and "or"-operations of fuzzy logic can be represented by the fastest (i.e., 1-layer) neural network? And a related question is: which activation functions allow such a representation? In this paper, we …

Commonsense Explanations Of Sparsity, Zipf Law, And Nash's Bargaining Solution, Olga Kosheleva, Vladik Kreinovich, Kittawit Autchariyapanitkul

Departmental Technical Reports (CS)

As econometric models become more and more accurate and more and more mathematically complex, they also become less and less intuitively clear and convincing. To make these models more convincing, it is desirable to supplement the corresponding mathematics with commonsense explanations. In this paper, we provide such explanation for three economics-related concepts: sparsity (as in LASSO), Zipf's Law, and Nash's bargaining solution.

Why Most Empirical Distributions Are Few-Modal, Julio Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In principle, any non-negative function can serve as a probability density function -- provided that it adds up to 1. All kinds of processes are possible, so it seems reasonable to expect that observed probability density functions are random with respect to some appropriate probability measure on the set of all such functions -- and for all such measures, similarly to the simplest case of random walk, almost all functions have infinitely many local maxima and minima. However, in practice, most empirical distributions have only a few local maxima and minima -- often one (unimodal distribution), sometimes two (bimodal), and, …

Why It Is Sufficient To Have Real-Valued Amplitudes In Quantum Computing, Isaac Bautista, Vladik Kreinovich, Olga Kosheleva, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

In the last decades, a lot of attention has been placed on quantum algorithms -- algorithms that will run on future quantum computers. In principle, quantum systems can use any complex-valued amplitudes. However, in practice, quantum algorithms only use real-valued amplitudes. In this paper, we provide a simple explanation for this empirical fact.

Optimization Under Fuzzy Constraints: Need To Go Beyond Bellman-Zadeh Approach And How It Is Related To Skewed Distributions, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

In many practical situations, we need to optimize the objective function under fuzzy constraints. Formulas for such optimization are known since the 1970s paper by Richard Bellman and Lotfi Zadeh, but these formulas have a limitation: small changes in the corresponding degrees can lead to a drastic change in the resulting selection. In this paper, we propose a natural modification of this formula, a modification that no longer has this limitation. Interestingly, this formula turns out to be related for formulas for skewed (asymmetric) generalizations of the normal distribution.

How To Estimate The Stiffness Of The Multi-Layer Road Based On Properties Of Layers: Symmetry-Based Explanation For Odemark's Equation, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich, Olga Kosheleva, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

When we design a road, we would like to check that the current design provides the pavement with sufficient stiffness to withstand traffic loads and climatic conditions. For this purpose, we need to estimate the stiffness of the road based on stiffness and thickness of its different layers. There exists a semi-empirical formula for this estimation. In this paper, we show that this formula can be explained by natural scale-invariance requirements.

How To Efficiently Store Intermediate Results In Quantum Computing: Theoretical Explanation Of The Current Algorithm, Oscar Galindo, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In complex time-consuming computations, we rarely have uninterrupted access to a high performance computer: usually, in the process of computation, some interruptions happen, so we need to store intermediate results until computations resume. To decrease the probability of a mistake, it is often necessary to run several identical computations in parallel, in which case several identical intermediate results need to be stored. In particular, for quantum computing, we need to store several independent identical copies of the corresponding qubits -- quantum versions of bits. Storing qubit states is not easy, but it is possible to compress the corresponding multi-qubit states: …

Why Linear Expressions In Discounting And In Empathy: A Symmetry-Based Explanation, Supanika Leurcharusmee, Laxman Bokati, Olga Kosheleva

Departmental Technical Reports (CS)

People's preferences depend not only on the decision maker's immediate gain, they are also affected by the decision maker's expectation of future gains. A person's decisions are also affected by possible consequences for others. In decision theory, people's preferences are described by special quantities called utilities. In utility terms, the above phenomena mean that the person's overall utility of an action depends not only on the utility corresponding to the action's immediate consequences for this person, it also depends on utilities corresponding to future consequences and on utilities corresponding to consequences for others. These dependencies reflect discounting of future consequences …

Is There A Contradiction Between Statistics And Fairness: From Intelligent Control To Explainable Ai, Christian Servin, Vladik Kreinovich

Departmental Technical Reports (CS)

At first glance, there seems to be a contradiction between statistics and fairness: statistics-based AI techniques lead to unfair discrimination based on gender, race, and socio-economical status. This is not just a fault of probability techniques: similar problems can happen if we use fuzzy or other techniques for processing uncertainty. To attain fairness, several authors proposed not to rely on statistics and instead, explicitly add fairness constraints into decision making. In this paper, we show that the seeming contradiction between statistics and fairness is caused mostly by the fact that the existing systems use simplified models; contradictions disappear if we …

Which Algorithms Are Feasible And Which Are Not: Fuzzy Techniques Can Help In Formalizing The Notion Of Feasibility, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Some algorithms are practically feasible, in the sense that for all inputs of reasonable length they provide the result in reasonable time. Other algorithms are not practically feasible, in the sense that they may work well for small-size inputs, but for slightly larger -- but still reasonable-size -- inputs, the computation time becomes astronomical (and not practically possible). How can we describe practical feasibility in precise terms? The usual formalization of the notion of feasibility states that an algorithm is feasible if its computation time is bounded by a polynomial of the size of the input. In most cases, this …

Centroids Beyond Defuzzification, Juan Carlos Figueroa-Garcia, Christian Servin, Vladik Kreinovich

Departmental Technical Reports (CS)

In general, expert rules expressed by imprecise (fuzzy) words of natural language like "small" lead to imprecise (fuzzy) control recommendations. If we want to design an automatic controller, we need, based on these fuzzy recommendations, to generate a single control value. A procedure for such generation is known as defuzzification. The most widely used defuzzification procedure is centroid defuzzification, in which, as the desired control value, we use one of the coordinates of the center of mass ("centroid") of an appropriate 2-D set. A natural question is: what is the meaning of the second coordinate of this center of mass? …

Optimal Search Under Constraints, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In general, if we know the values a and b at which a continuous function has different signs -- and the function is given as a black box -- the fastest possible way to find the root x for which f(x) = 0 is by using bisection (also known as binary search). In some applications, however -- e.g., in finding the optimal dose of a medicine -- we sometimes cannot use this algorithm since, for avoid negative side effects, we can only try value which exceed the optimal dose by no more than some small value δ > 0. In this …

Equations For Which Newton's Method Never Works: Pedagogical Examples, Leobardo Valera, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the most widely used methods for solving equations is the classical Newton's method. While this method often works -- and is used in computers for computations ranging from square root to division -- sometimes, this method does not work. Usual textbook examples describe situations when Newton's method works for some initial values but not for others. A natural question that students often ask is whether there exist functions for which Newton's method never works -- unless, of course, the initial approximation is already the desired solution. In this paper, we provide simple examples of such functions.

Scale-Invariance Ideas Explain The Empirical Soil-Water Characteristic Curve, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

The prediction of the road's properties under the influence of water infiltration is important for pavement design and management. Traditionally, this prediction heavily relied on expert estimates. In the last decades, complex empirical formulas have been proposed to capture the expert's intuition in estimating the effect of water infiltration on the stiffness of the pavement's payers. Of special importance is the effect of water intrusion on the pavement's foundation -- known as subgrade soil. In this paper, we show that natural scale-invariance ideas lead to a theoretical explanation for an empirical formula describing the dependence between soil suction and water …

Why There Are Only Four Fundamental Forces: A Possible Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that there are exactly four fundamental forces of nature: gravity forces, forces corresponding to weak interactions, electromagnetic forces, and forces corresponding to strong interactions. In this paper, we provide a possible explanation of why there are exactly four fundamental forces: namely, we relate this number with the dimension of physical space-time.