Computer Sciences Commons^™

Unreachable Statements Are Inevitable In Software Testing: Theoretical Explanation, Francisco Zapata, Eric Smith, Vladik Kreinovich

Departmental Technical Reports (CS)

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 surprisingly, for many companies, the mere presence of such aspirational goals boosts the company's performance. In this paper, we show that a simple geometric model of this situation can explain the unexpected success of aspirational goals.

A Natural Causality-Motivated Description Of Learning, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Teaching is not easy. One of the main reasons why it is not easy is that the existing descriptions of the teaching process are not very precise -- and thus, we cannot use the usual optimization techniques, techniques which require a precise model of the corresponding phenomenon. It is therefore desirable to come up with a precise description of the learning process. To come up with such a description, we notice that on the set of all possible states of learning, there is a natural order s ≤ s' meaning that we can bring the student from the state s …

Why Gaussian Copulas Are Ubiquitous In Economics: Fuzzy-Related Explanation, Chon Van Le, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many real-life situations, deviations are caused by a large number of independent factors. It is known that in such situations, the distribution of the resulting deviations is close to Gaussian, and thus, that the copulas -- that describe the multi-D distributions as a function of 1-D (marginal) ones -- are also Gaussian. In the past, these conclusions were also applied to economic phenomena, until the 2008 crisis showed that in economics, Gaussian models can lead to disastrous consequences. At present, all economists agree that the economic distributions are not Gaussian -- however, surprisingly, Gaussian copulas still often provide an …

Video Or Text? Bullets Or No Bullets? Why Not Both?, Olga Kosheleva, Vladik Kreinovich, Christian Servin

Departmental Technical Reports (CS)

Some students – which are, in terms of pop-psychology – more left-brain – prefer linear exposition, others – more right-brain ones – prefer 2-D images and texts with visual emphasis (e.g., with bullets). At present, instructors try to find a middle grounds between these two audiences, but why not prepare each material in two ways, aimed at both audiences?

Computing The Range Of A Function-Of-Few-Linear-Combinations Under Linear Constraints: A Feasible Algorithm, Salvador Robles, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we need to find the range of a given function under interval uncertainty. For nonlinear functions -- even for quadratic ones -- this problem is, in general, NP-hard; however, feasible algorithms exist for many specific cases. In particular, recently a feasible algorithm was developed for computing the range of the absolute value of a Fourier coefficient under uncertainty. In this paper, we generalize this algorithm to the case when we have a function of a few linear combinations of inputs. The resulting algorithm also handles the case when, in addition to intervals containing each input, we …

Commonsense-Continuous Dynamical Systems -- Stationary States, Prediction, And Reconstruction Of The Past: Fuzzy-Based Analysis, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditional analysis of dynamical systems usually assumes that the mapping is continuous -- in precise mathematical sense. However, as many formal definitions, the mathematical definition of continuity does not always adequately capture the commonsense notion of continuity: that small changes in the input should lead to small changes in the output. In this paper, we provide a natural fuzzy-based formalization of this intuitive notion, and analyze how the requirement of commonsense continuity affects the properties of dynamical systems. Specifically, we show that for such systems, the set of fixed points is closed and convex, and that the only such systems …

Need For Techniques Intermediate Between Interval And Probabilistic Ones, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

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 in which all measurement errors are strongly correlated, and the probabilistic approach, when we assume that all these errors are independent. The problem is that usually, the interval approach leads to too pessimistic, too large uncertainty estimates, while the probabilistic approach often underestimates the resulting uncertainty. To get …

Fuzzy Or Neural, Type-1 Or Type-2 -- When Each Is Better: First-Approximation Analysis, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

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 work better. Usually, practitioners try several techniques and select the one that works best for their problem. This trying often requires a lot of efforts. It would be more efficient if we could have a priori recommendations about which technique is better. In this paper, we use the first-approximation …

Why Ideas First Appear In Informal Form? Why It Is Very Difficult To Know Yourself? Fuzzy-Based Explanation, Miroslav Svitek, Vladik Kreinovich

Departmental Technical Reports (CS)

To a lay person reading about history of physics, it may sound as if the progress of physics comes from geniuses whose inspiration leads them to precise equations that -- almost magically -- explain all the data: this is what Newton did with mechanics, this is what Schroedinger did with quantum physics, this is what Einstein did with gravitation. However, a deeper study of history of physics shows that in all these cases, these geniuses did not start from scratch -- they formalized ideas that first appeared in imprecise ("fuzzy") form. In this paper, we explain -- on the qualitative …

Why Aspirational Goals: Geometric Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

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 surprisingly, for many companies, the mere presence of such aspirational goals boosts the company's performance. In this paper, we show that a simple geometric model of this situation can explain the unexpected success of aspirational goals.

Why Pre-Teaching: A Geometric Explanation, Olga Kosheleva, Vladik Kreinovich, Christian Servin

Departmental Technical Reports (CS)

Traditionally, subjects are taught in sequential order: e.g., first, students study algebra, then they use the knowledge of algebra to study the basis ideas of calculus. In this traditional scheme, teachers usually do not explain any calculus ideas before students are ready – since they believe that this would only confuse students. However, lately, empirical evidence has shows that, contrary to this common belief, pre-teaching – when students get a brief introduction to the forthcoming new topic before this topic starts – helps students learn. In this paper, we provide a geometric explanation for this unexpected empirical phenomenon.

Why Core Curriculum? Why Art And Nature Enhance Creativity? A Mathematical Explanation, Olga Kosheleva, Vladik Kreinovich, Christian Servin

Departmental Technical Reports (CS)

Teaching is not easy. One of the main reasons why it is not easy is that the existing descriptions of the teaching process are not very precise -- and thus, we cannot use the usual optimization techniques, techniques which require a precise model of the corresponding phenomenon. It is therefore desirable to come up with a precise description of the learning process. To come up with such a description, we notice that on the set of all possible states of learning, there is a natural order s ≤ s' meaning that we can bring the student from the state s …

Data Processing Under Fuzzy Uncertainty: Towards More Efficient Algorithm, Hung T. Nguyen, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we need to process data under fuzzy uncertainty: we have fuzzy information about the algorithm's input, and we want to find the resulting information about the algorithm's output. It is known that this problem can be reduced to computing the range of the algorithm over alpha-cuts of the input. Since the fuzzy degrees are usually known with accuracy at best 0.1, it is sufficient to repeat this range-computing procedure for 11 values alpha = 0, 0.1, ..., 1.0. However, a straightforward application of this idea requires 11 times longer computation time than each range estimation -- …

How To Describe Relative Approximation Error? A New Justification For Gustafson's Logarithmic Expression, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

How can we describe relative approximation error? When the value b approximate a value a, the usual description of this error is the ratio |b − a|/|a|. The problem with this approach is that, contrary to our intuition, we get different numbers gauging how well a approximates b and how well b approximates a. To avoid this problem, John Gustafson proposed to use the logarithmic measure |ln(b/a)|. In this paper, we show that this is, in effect, the only regular scale-invariant way to describe the relative approximation error.

How To React To Student Evaluations, Olga Kosheleva, Vladik Kreinovich, Christian Servin

Departmental Technical Reports (CS)

If most students comment that the course was too fast, a natural idea is to slow it down. If most students comment that the course was too slow, a natural idea is to speed it up. But what if half the students think the speed was too fast and half that the speed was too slow? A frequent reaction to such a situation is to conclude that the speed was just right and not change the speed the next time, but this may not be the right reaction: under the same speed, half of the students will struggle and may …

Provably Weak Instances Of Plwe Revisited, Again, Katherine Mendel

CSB and SJU Distinguished Thesis

Learning with Errors has emerged as a promising possibility for postquantum cryptography. Variants known as RLWE and PLWE have been shown to be more efficient, but the increased structure can leave them vulnerable to attacks for certain instantiations. This work aims to identify specific cases where proposed cryptographic schemes based on PLWE work particularly poorly under a specific attack.

The Nature Of Numbers: Real Computing, Bradley J. Lucier

Journal of Humanistic Mathematics

While studying the computable real numbers as a professional mathematician, I came to see the computable reals, and not the real numbers as usually presented in undergraduate real analysis classes, as the natural culmination of my evolving understanding of numbers as a schoolchild. This paper attempts to trace and explain that evolution. The first part recounts the nature of numbers as they were presented to us grade-school children. In particular, the introduction of square roots induced a step change in my understanding of numbers. Another incident gave me insight into the brilliance of Alan Turing in his paper introducing both …

Optimal Eavesdropping In Quantum Cryptography, Atanu Acharyya Dr.

Doctoral Theses

Quantum key distribution (QKD) has raised some promise for more secured communication than its classical counterpart. It allows the legitimate parties to detect eavesdropping which introduces error in the channel. If disturbed, there are ways to distill a secure key within some threshold error-rate. The amount of information gained by an attacker is generally quantified by (Shannon) mutual information. Knowing the maximum amount of information that an intruder can gain is important for post-processing purposes, and we mainly focus on that side in the thesis. Renyi information is also useful especially when post-processing is considered. The scope of this thesis …

Why Sine Membership Functions, Sofia Holguin, Javier Viaña, Kelly Cohen, Anca Ralescu, Vladik Kreinovich

Departmental Technical Reports (CS)

In applications of fuzzy techniques to several practical problems -- in particular, to the problem of predicting passenger flows in the airports -- the most efficient membership function is a sine function; to be precise, a portion of a sine function between the two zeros. In this paper, we provide a theoretical explanation for this empirical success.

Need To Combine Interval And Probabilistic Uncertainty: What Needs To Be Computed, What Can Be Computed, What Can Be Feasibly Computed, And How Physics Can Help, Julio Urenda, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

In many practical situations, the quantity of interest is difficult to measure directly. In such situations, to estimate this quantity, we measure easier-to-measure quantities which are related to the desired one by a known relation, and we use the results of these measurement to estimate the desired quantity. How accurate is this estimate?

Traditional engineering approach assumes that we know the probability distributions of measurement errors; however, in practice, we often only have partial information about these distributions. In some cases, we only know the upper bounds on the measurement errors; in such cases, the only thing we know about …

Why People Tend To Overestimate Joint Probabilities, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that, in general, people overestimate the probabilities of joint events. In this paper, we provide an explanation for this phenomenon -- as explanation based on Laplace Indeterminacy Principle and Maximum Entropy approach.

How To Deal With Conflict Of Interest Situations When Selecting The Best Submission, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

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 can we take these opinions into account? In this paper, we show that a seemingly reasonable idea can actually lead to bias, and we explain how to take these opinions into account without biasing the final …

What Is A Natural Probability Distribution On The Class Of All Continuous Functions: Maximum Entropy Approach Leads To Wiener Measure, Vladik Kreinovich, Saeid Tizpaz-Niari

Departmental Technical Reports (CS)

While many data processing techniques assume that we know the probability distributions, in practice, we often only have a partial information about these probabilities -- so that several different distributions are consistent with our knowledge. Thus, to apply these data processing techniques, we need to select one of the possible probability distributions. There is a reasonable approach for such selection -- the Maximum Entropy approach. This approach selects a uniform distribution if all we know is that the random variable if located in an interval; it selects a normal distribution if all we know is the mean and the variance. …

Macrocausality Implies Lorenz Group: A Physics-Related Comment On Guts's Results, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that, in the space-time of Special Relativity, causality implies Lorenz group, i.e., if we know which events can causally influence each other, then, based on this information, we can uniquely reconstruct the affine structure of space-time. When the two events are very close, quantum effects, with their probabilistic nature, make it difficult to detect causality. So, the following question naturally arises: can we uniquely reconstruct the affine structure if we only know causality for events which are sufficiently far away from each other? Several positive answers to this question were provided in a recent paper by Alexander …

Local-Global Results On Discrete Structures, Alexander Lewis Stevens

Electronic Theses and Dissertations

Local-global arguments, or those which glean global insights from local information, are central ideas in many areas of mathematics and computer science. For instance, in computer science a greedy algorithm makes locally optimal choices that are guaranteed to be consistent with a globally optimal solution. On the mathematical end, global information on Riemannian manifolds is often implied by (local) curvature lower bounds. Discrete notions of graph curvature have recently emerged, allowing ideas pioneered in Riemannian geometry to be extended to the discrete setting. Bakry- Émery curvature has been one such successful notion of curvature. In this thesis we use combinatorial …

Guide To The Dr. L.S. Dederick Papers, 1908-1956, Undated, Orson Kingsley, Patrick Koetsch

Archives & Special Collections Finding Aids

Louis Serle (L.S.) Dederick was born in Chicago in 1883. He received his Ph.D. in Mathematics from Harvard University in 1909. From 1909 – 1917 he was a professor at Princeton University. From 1917 – 1924 he was professor at the U.S. Naval Academy in Annapolis, Maryland. In 1926 Dederick began working for the U.S. Army, Ordnance. During his time there he was the Associate Director of the Ballistic Research Laboratory at the Aberdeen Proving Grounds in Aberdeen, Maryland where he focused on ballistics research.

While Dederick worked as a mathematician at the Aberdeen Proving Grounds, he was involved with …

A Predictive Model To Predict Cyberattack Using Self-Normalizing Neural Networks, Oluwapelumi Eniodunmo

Theses, Dissertations and Capstones

Cyberattack is a never-ending war that has greatly threatened secured information systems. The development of automated and intelligent systems provides more computing power to hackers to steal information, destroy data or system resources, and has raised global security issues. Statistical and Data mining tools have received continuous research and improvements. These tools have been adopted to create sophisticated intrusion detection systems that help information systems mitigate and defend against cyberattacks. However, the advancement in technology and accessibility of information makes more identifiable elements that can be used to gain unauthorized access to systems and resources. Data mining and classification tools …

Search Under Uncertainty Should Be Randomized: A Lesson From The 2021 Nobel Prize In Medicine, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

In many real-life situations, we know that one of several objects has the desired property, but we do not know which one. To find the desired object, we need to test these objects one by one. In situations when we have no additional information, there is no reason to prefer any testing order and thus, a usual recommendation is to test them in any order. This is usually interpreted as ordering the objects in the increasing value of some seemingly unrelated quantity. A possible drawback of this approach is that it may turn out that the selected quantity is correlated …

Games For One, Games For Two: Computationally Complex Fun For Polynomial-Hierarchical Families, Kye Shi

HMC Senior Theses

In the first half of this thesis, we explore the polynomial-time hierarchy, emphasizing an intuitive perspective that associates decision problems in the polynomial hierarchy to combinatorial games with fixed numbers of turns. Specifically, problems in �� are thought of as 0-turn games, ���� as 1-turn “puzzle” games, and in general ��ₖ�� as ��-turn games, in which decision problems answer the binary question, “can the starting player guarantee a win?” We introduce the formalisms of the polynomial hierarchy through this perspective, alongside definitions of ��-turn CIRCUIT SATISFIABILITY games, whose ��ₖ��-completeness is assumed from prior work (we briefly justify this assumption …

Finding Optimal Cayley Map Embeddings Using Genetic Algorithms, Jacob Buckelew

Honors Program Theses

Genetic algorithms are a commonly used metaheuristic search method aimed at solving complex optimization problems in a variety of fields. These types of algorithms lend themselves to problems that can incorporate stochastic elements, which allows for a wider search across a search space. However, the nature of the genetic algorithm can often cause challenges regarding time-consumption. Although the genetic algorithm may be widely applicable to various domains, it is not guaranteed that the algorithm will outperform other traditional search methods in solving problems specific to particular domains. In this paper, we test the feasibility of genetic algorithms in solving a …