Computer Sciences Commons^™

Need For Diversity In Elected Decision-Making Bodies: Economics-Related Analysis, Nguyen Ngoc Thach, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

On a qualitative level, everyone understands the need to have diversity in elected decision-making bodies, so that the viewpoint of each group be properly taken into account. However, when only the usual economic criteria are used in this election -- e.g., in the election of company's board -- the resulting bodies often under-represent some groups (e.g., women). A frequent way to remedy this situation is to artificially enforce diversity instead of strictly following purely economic criteria. In this paper, we show the current seeming contradiction between economics and diversity is caused by the imperfection of the use economic models: in …

Grading Homeworks, Verifying Code: How Thorough Should The Feedback Be?, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the ideal world, we should assign many homeworks and give a thorough feedback for each homework. However, in reality, the instructor's time is limited, so we can either assign few homeworks and give a detailed feed back for all of them, or we can assign many homeworks, but give a less thorough feedback. What is the optimal thoroughness? A similar question can be raised for code verification: what is the optimal amount of feedback that should be provided to each programmer? In this paper, we provide answers to these questions.

Why Majority Rule Does Not Work In Quantum Computing: A Pedagogical Explanation, Oscar Galindo, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

To increase the reliability of computations result, a natural idea is to use duplication: we let several computers independently perform the same computations, and then, if their results differ, we select the majority's result. Reliability is an important issue for quantum computing as well, since in quantum physics, all the processes are probabilistic, so there is always a probability that the result will be wrong. It thus seems natural to use the same majority rule for quantum computing as well. However, it is known that for general quantum computing, this scheme does not work. In this paper, we provide a …

Adversarial Teaching Approach To Cybersecurity: A Mathematical Model Explains Why It Works Well, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Teaching cybersecurity means teaching all possible ways how software can be attacked -- and how to fight such attacks. From the usual pedagogical viewpoint, a natural idea seems to be to teach all these ways one by one. Surprisingly, a completely different approach works even better: when the class is divided into sparring mini-teams that try their best to attack each other and defend from each other. In spite of the lack of thoroughness, this approach generates good specialists -- but why? In this paper, by analyzing a simple mathematical model of this situation, we explain why this approach work …

Let Us Use Negative Examples In Regression-Type Problems Too, Jonatan Contreras, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich, Martine Ceberio

Departmental Technical Reports (CS)

In many practical situations, we need to reconstruct the dependence between quantities x and y based on several situations in which we know both x and y values. Such problems are known as regression problems. Usually, this reconstruction is based on positive examples, when we know y -- at least, with some accuracy. However, in addition, we often also know some examples in which we have negative information about y -- e.g., we know that y does not belong to a certain interval. In this paper, we show how such negative examples can be used to make the solution …

How To Make Sure That Robot's Behavior Is Human-Like, Vladik Kreinovich, Olga Kosheleva, Laxman Bokati

Departmental Technical Reports (CS)

In many applications -- e.g., in health care -- it is desirable to make robots behave human-like. This means, in particular, that robotic control should not be optimal, it should be similar to human (suboptimal) behavior. People's decisions are based on bounded rationality: since we cannot compute an optimal solution for all possible situations, we divide situations into groups and come up with a solution appropriate for each group. What is optimal here is the division into groups. It is therefore desirable to implement a similar algorithm for robots. To help with such algorithms, we provide techniques that help optimally …

Are There Traces Of Megacomputing In Our Universe, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

The recent successes of quantum computing encouraged many researchers to search for other unconventional physical phenomena that could potentially speed up computations. Several promising schemes have been proposed that will -- hopefully -- lead to faster computations in the future. Some of these schemes -- similarly to quantum computing -- involve using events from the micro-world, others involve using large-scale phenomena. If some civilization used micro-world for computations, this will be difficult for us to notice, but if they use mega-scale effects, maybe we can notice these phenomena? In this paper, we analyze what possible traces such megacomputing can leave …

How Mathematics And Computing Can Help Fight The Pandemic: Two Pedagogical Examples, Julio Urenda, Olga Kosheleva, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

With the 2020 pandemic came unexpected mathematical and computational problems. In this paper, we provide two examples of such problems -- examples that we present in simplified pedagogical form. The problems are related to the need for social distancing and to the need for fast testing. We hope that these examples will help students better understand the importance of mathematical models.

Natural Invariance Explains Empirical Success Of Specific Membership Functions, Hedge Operations, And Negation Operations, Julio Urenda, Orsoly Csiszár, Gábor Csiszár, József Dombi, György Eigner, Vladik Kreinovich

Departmental Technical Reports (CS)

Empirical studies have shown that in many practical problems, out of all symmetric membership functions, special distending functions work best, and out of all hedge operations and negation operations, fractional linear ones work the best. In this paper, we show that these empirical successes can be explained by natural invariance requirements.

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 …

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.

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 …

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 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: …

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 …

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.

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.

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.

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.

Why Black-Scholes Equations Are Effective Beyond Their Usual Assumptions: Symmetry-Based Explanation, Warattaya Chinnakum, Sean R. Aguilar

Departmental Technical Reports (CS)

Nobel-Prize-winning Black-Scholes equations are actively used to estimate the price of options and other financial instruments. In practice, they provide a good estimate for the price, but the problem is that their original derivation is based on many simplifying statistical assumptions which are, in general, not valid for financial time series. The fact that these equations are effective way beyond their usual assumptions leads to a natural conclusion that there must be an alternative derivation for these equations, a derivation that does not use the usual too-strong assumptions. In this paper, we provide such a derivation in which the only …

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? …

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 …

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 …

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 …

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 …

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.

Why Class-D Audio Amplifiers Work Well: A Theoretical Explanation, Kevin Alvarez, Julio Urenda, Vladik Kreinovich

Departmental Technical Reports (CS)

Most current high-quality electronic audio systems use class-D audio amplifiers (D-amps, for short), in which a signal is represented by a sequence of pulses of fixed height, pulses whose duration at any given moment of time linearly depends on the amplitude of the input signal at this moment of time. In this paper, we explain the efficiency of this signal representation by showing that this representation is the least vulnerable to additive noise (that affect measuring the signal itself) and to measurement errors corresponding to measuring time.