Physical Sciences and Mathematics Commons^™

From Fuzzy Universal Approximation To Fuzzy Universal Representation: It All Depends On The Continuum Hypothesis, Mahdokhat Michelle Afravi, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that fuzzy systems have a universal approximation property. A natural question is: can this property be extended to a universal representation property? Somewhat surprisingly, the answer to this question depends on whether the following Continuum Hypothesis: every infinite subset of the real line has either the same number of elements as the real line itself or as many elements as natural numbers.

Isn't Every Sufficiently Complex Logic Multi-Valued Already: Lindenbaum-Tarski Algebra And Fuzzy Logic Are Both Particular Cases Of The Same Idea, Andrzej Pownuk, Vladik Kreinovich

Departmental Technical Reports (CS)

Usually, fuzzy logic (and multi-valued logics in general) are viewed as drastically different from the usual 2-valued logic. In this paper, we show that while on the surface, there indeed seems to be a major difference, a more detailed analysis shows that even in the theories based on the 2-valued logic, there naturally appear constructions which are, in effect, multi-valued, constructions which are very close to fuzzy logic.

No Idea Is A Bad Idea: A Theoretical Explanation, Christian Servin, Vladik Kreinovich

Departmental Technical Reports (CS)

Many business publications state that no idea is a bad idea, that even if the idea is, at first glance, not helpful, there are usually some aspects of this idea which are helpful Usually, this statement is based on the experience of the author, and it is given without any theoretical explanation. In this paper, we provide a theoretical explanation for this statement.

Maybe The Usual Students' Practice Of Cramming For A Test Makes Sense: A Mathematical Analysis, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

We always teach students that cramming for a test is a bad idea, that they should study at the same speed throughout the semester – but many still cram. We ourselves are not that different: when we prepare papers for a conference, we often “cram” in the last days before the deadline instead of working with a regular speed for the whole time before the conference. The ubiquity of cramming makes us think that maybe it is not necessarily always a bad idea. And indeed, a simple model of a study process shows that an optimal solution often involve some …

A Natural Feasible Algorithm That Checks Satisfiability Of 2-Cnf Formulas And, If The Formulas Is Satisfiable, Finds A Satisfying Vector, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the main results in Theory of Computation courses is the proof that propositional satisfiability is NP-complete. This means that, unless P = NP (which most computer scientists believe to be impossible), no feasible algorithm is possible for solving propositional satisfiability problems. This result is usually proved on the example of 3-CNF formulas, i.e., formulas of the type C₁ & ... & C_m, where each clause C_i has the form a \/ b or a \/ b \/ c, with no more than three literals -- i.e., propositional variables v_i or their negations ~v …

What Decision To Make In A Conflict Situation Under Interval Uncertainty: Efficient Algorithms For The Hurwzcz Approach, Bartłomiej Jacek Kubica, Andrzej Pownuk, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we show how to take interval uncertainty into account when solving conflict situations. Algorithms for conflict situations under interval uncertainty are know under the assumption that each side of the conflict maximizes its worst-case expected gain. However, it is known that a more general Hurwicz approach provides a more adequate description of decision making under uncertainty. In this approach, each side maximizes the convex combination of the worst-case and the best-case expected gains. In this paper, we describe how to resolve conflict situations under the general Hurwicz approach to interval uncertainty.

Fuzzy Sets As Strongly Consistent Random Sets, Kittawit Autchariyapanitkul, Hung T. Nguyen, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that from the purely mathematical viewpoint, fuzzy sets can be interpreted as equivalent classes of random sets. This interpretations helps to teach fuzzy techniques to statisticians and also enables us to apply results about random sets to fuzzy techniques. The problem with this interpretation is that it is too complicated: a random set is not an easy notion, and classes of random sets are even more complex. This complexity goes against the spirit of fuzzy sets, whose purpose was to be simple and intuitively clear. From this viewpoint, it is desirable to simplify this interpretation. In this …

Attraction-Repulsion Forces Between Biological Cells: A Theoretical Explanation Of Empirical Formulas, Olga Kosheleva, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

Biological calls attract and repulse each other: if they get too close to each other, they repulse, and if they get too far away from each other, they attract. There are empirical formulas that describe the dependence of the corresponding forces on the distance between the cells. In this paper, we provide a theoretical explanation for these empirical formulas.

How To Teach Implication, Martha Osegueda Escobar, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Logical implication is a somewhat counter-intuitive notion. For students, it is difficult to understand why a false statement implies everything. In this paper, we present a simple pedagogical way to make logical implication more intuitive.

Fuzzy Techniques Explain Empirical Power Law Governing Wars And Terrorist Attacks, Hung T. Nguyen, Kittawit Autchariyapanitkul, Vladik Kreinovich

Departmental Technical Reports (CS)

The empirical distribution of the number of casualties in wars and terrorist attacks follows a power law with exponent 2.5. So far, there has not been a convincing explanation for this empirical fact. In this paper, we show that by using fuzzy techniques, we can explain this exponent. Interesting, we can also get a similar explanation if we use probabilistic techniques. The fact that two different techniques lead to the same explanation makes us reasonably confident that this explanation is correct.

Uncertain Information Fusion And Knowledge Integration: How To Take Reliability Into Account, Hung T. Nguyen, Kittawit Autchariyapanitkul, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we need to fuse and integrate information and knowledge from different sources -- and do it under uncertainty. Most existing methods for information fusion and knowledge integration take into account uncertainty. In addition to uncertainty, we also face the problem of reliability: sensors may malfunction, experts can be wrong, etc. In this paper, we show how to take into account both uncertainty and reliability in information fusion and knowledge integration. We show this on the examples of probabilistic and fuzzy uncertainty.

Which Material Design Is Possible Under Additive Manufacturing: A Fuzzy Approach, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Additive manufacturing -- also known as 3-D printing -- is a very promising new way to generate complex material designs. However, even with the modern advanced techniques, some designs are too complex to be implemented. There exist an empirical formula that describes when the design is implementable. In this paper, we use fuzzy ideas to provide a theoretical justification for this empirical formula.

It Is Possible To Determine Exact Fuzzy Values Based On An Ordering Of Interval-Valued Or Set-Valued Fuzzy Degrees, Gerardo Muela, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the usual [0,1]-based fuzzy logic, the actual numerical value of a fuzzy degree can be different depending on a scale, what is important -- and scale-independent -- is the order between different values. To make a description of fuzziness more adequate, it is reasonable to consider interval-valued degrees instead of numerical ones. Here also, what is most important is the order between the degrees. If we have only order between the intervals, can we, based on this order, reconstruct the original numerical values -- i.e., the degenerate intervals? In this paper, we show that such a reconstruction is indeed …

A Short Note On Pitch, Interval, And Melody Matching Assessment, Eric Hanson, Hannah Baslee, Eric Freudenthal

Departmental Technical Reports (CS)

This short note describes a metric and procedure for assessing an individual's overall simple pitch and interval matching proficiency when singing.

A Symmetry-Based Explanation For An Empirical Model Of Fatigue Damage Of Composite Materials, Pedro Barragan Olague, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we provide a symmetry-based explanation for an empirical formula that describes fatigue damage of composite materials.

Prediction Of Volcanic Eruptions As A Case Study Of Predicting Rare Events In Chaotic Systems With Delay, Justin Parra, Olac Fuentes, Elizabeth Y. Anthony, Vladik Kreinovich

Departmental Technical Reports (CS)

Volcanic eruptions can be disastrous; it is therefore important to be able to predict them as accurately as possible. Theoretically, we can use the general machine learning techniques for such predictions. However, in general, without any prior information, such methods require an unrealistic amount of computation time. It is therefore desirable to look for additional information that would enable us to speed up the corresponding computations. In this paper, we provide an empirical evidence that the volcanic system exhibit chaotic and delayed character. We also show that in general (and in volcanic predictions in particular), we can speed up the …

In System Identification, Interval (And Fuzzy) Estimates Can Lead To Much Better Accuracy Than The Traditional Statistical Ones: General Algorithm And Case Study, Sergey I. Kumkov, Vladik Kreinovich, Andrzej Pownuk

Departmental Technical Reports (CS)

In many real-life situations, we know the upper bound of the measurement errors, and we also know that the measurement error is the joint result of several independent small effects. In such cases, due to the Central Limit theorem, the corresponding probability distribution is close to Gaussian, so it seems reasonable to apply the standard Gaussian-based statistical techniques to process this data -- in particular, when we need to identify a system. Yes, in doing this, we ignore the information about the bounds, but since the probability of exceeding them is small, we do not expect this to make a …

The Onsager Conjecture: A Pedagogical Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In 1949, a Nobelist Lars Onsager considered liquid flows with velocities changing as r^α for spatial points at distance r, and conjectured that the threshold value α = 1/3 separates the two possible regimes: for α > 1/3 energy is always preserved, while for α < 1/3 energy is possibly not preserved. In this paper, we provide a simple pedagogical explanation for this conjecture.

Soft Computing Approach To Detecting Discontinuities: Seismic Analysis And Beyond, Solymar Ayala Cortez, Aaron A. Velasco, Vladik Kreinovich

Departmental Technical Reports (CS)

Starting from Newton, the main equations of physics are differential equations -- which implicitly implies that all the corresponding processes are differentiable -- and thus, continuous. However, in practice, we often encounter processes or objects that change abruptly in time or in space. In physics, we have phase transitions when the properties change abruptly. In geosciences, we have sharp boundaries between different layers and discontinuing representing faults. In many such situations, it is important to detect these discontinuities. In some cases, we know the equations, but in many other cases, we do not know the equations, we only know that …

Towards Decision Making Under General Uncertainty, Andrzej Pownuk, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

There exist techniques for decision making under specific types of uncertainty, such as probabilistic, fuzzy, etc. Each of the corresponding ways of describing uncertainty has its advantages and limitations. As a result, new techniques for describing uncertainty appear all the time. Instead of trying to extend the existing decision making idea to each of these new techniques one by one, we attempt to develop a general approach that would cover all possible uncertainty techniques.

Can We Detect Crisp Sets Based Only On The Subsethood Ordering Of Fuzzy Sets? Fuzzy Sets And/Or Crisp Sets Based On Subsethood Of Interval-Valued Fuzzy Sets?, Christian Servin, Gerardo Muela, Vladik Kreinovich

Departmental Technical Reports (CS)

Fuzzy sets are naturally ordered by the subsethood relation. If we only know which set which fuzzy set is a subset of which -- and have no access to the actual values of the corresponding membership functions -- can we detect which fuzzy sets are crisp? In this paper, we show that this is indeed possible. We also show that if we start with interval-valued fuzzy sets, then we can similarly detect type-1 fuzzy sets and crisp sets.

Why Convex Optimization Is Ubiquitous And Why Pessimism Is Widely Spread, Angel F. Garcia Contreras, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical applications, the objective function is convex. The use of convex objective functions makes optimization easier, but ubiquity of such objective function is a mystery: many practical optimization problems are not easy to solve, so it is not clear why the objective function -- whose main goal is to describe our needs -- would always describe easier-to-achieve goals. In this paper, we explain this ubiquity based on the fundamental ideas about human decision making. This explanation also helps us explain why in decision making under uncertainty, people often make pessimistic decisions, i.e.., decisions based on the worst-case scenarios.

How To Deal With Uncertainties In Computing: From Probabilistic And Interval Uncertainty To Combination Of Different Approaches, With Applications To Engineering And Bioinformatics, Vladik Kreinovich

Departmental Technical Reports (CS)

Most data processing techniques traditionally used in scientific and engineering practice are statistical. These techniques are based on the assumption that we know the probability distributions of measurement errors etc.

In practice, often, we do not know the distributions, we only know the bound D on the measurement accuracy -- hence, after the get the measurement result X, the only information that we have about the actual (unknown) value x of the measured quantity is that $x$ belongs to the interval [X − D, X + D]. Techniques for data processing under such interval uncertainty are called interval computations; these …

Why Stable Teams Are More Efficient In Education, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that study groups speed up learning. Recent studies have shown that stable study groups are more efficient than shifting-membership groups. In this paper, we provide a theoretical explanation for this empirical observation.

Contradictions Do Not Necessarily Make A Theory Inconsistent, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Some religious scholars claim that while the corresponding holy texts may be contradictory, they lead to a consistent set of ethical and behavioral recommendations. Is this logically possible? In this paper, somewhat surprisingly, we kind of show that this is indeed possible: namely, we show that if we add, to statements about objects from a certain class, consequences of both contradictory abstract statements, we still retain a consistent theory. A more mundane example of the same phenomenon comes from mathematics: if we have a set-theoretical statement S which is independent from ZF and which is not equivalent to any arithmetic …

Towards Predictive Statistics: A Pedagogical Explanation, Vladik Kreinovich

Departmental Technical Reports (CS)

In statistics application area, lately, several publications appeared that warn about the dangers of the inappropriate application of statistics and remind the users of the recall that prediction is the ultimate objective of the statistical analysis. This trend is known as predictive statistics. However, while the intended message is aimed at the very general audience of practitioners and researchers who apply statistics, many of these papers are not easy to read since they are either too technical and/or too philosophical for the general reader. In this short paper, we describe the main ideas and recommendation of predictive statistics in …

Why Are Fgm Copulas Successful: A Simple Explanation, Songsak Sriboonchitta, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the most computationally convenient non-redundant ways to describe the dependence between two variables is by describing the corresponding copula. In many application, a special class of copulas -- known as FGM copulas -- turned out to be most successful in describing the dependence between quantities. The main result of this paper is that these copulas are the fastest-to-compute, and this explains their empirical success.

As an auxiliary result, we also show that a similar explanation can be given in terms of fuzzy logic.

Physical Induction Explains Why Over-Realistic Animation Sometimes Feels Creepy, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the past, every progress of movie animation towards realism was viewed positively. However, recently, as computer animation is becoming more and more realistic, some people perceive the resulting realism negatively, as creepy. Similarly, everyone used to welcome robots that looked and behaved somewhat like humans; however, lately, too-human-like robots have started causing a similar negative feeling of creepiness. There exist complex psychology-based explanations for this phenomenon. In this paper, we show that this empirical phenomenon can be naturally explained simply by physical induction -- the main way we cognize the world.

Why Linear Interpolation?, Andrzej Pownuk, Vladik Kreinovich

Departmental Technical Reports (CS)

Linear interpolation is the computationally simplest of all possible interpolation techniques. Interestingly, it works reasonably well in many practical situations, even in situations when the corresponding computational models are rather complex. In this paper, we explain this empirical fact by showing that linear interpolation is the only interpolation procedure that satisfies several reasonable properties such as consistency and scale-invariance.

Derivation Of Gross-Pitaevskii Version Of Nonlinear Schroedinger Equation From Scale Invariance, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that in the usual 3-D space, the Schroedinger equation can be derived from scale-invariance. In view of the fact that, according to modern physics, the actual dimension of proper space may be different from 3, it is desirable to analyze what happens in other spatial dimensions D. It turns out that while for D ≥ 3 we still get only the Schroedinger's equation, for D = 2, we also get the Gross-Pitaevskii version of a nonlinear Schroedinger equation that describes a quantum system of identical bosons, and for D = 1, we also get a new nonlinear …