Data-driven software is increasingly being used as a critical component of automated decision-support systems. Since this class of software learns its logic from historical data, it can encode or amplify discriminatory practices. Previous research on algorithmic fairness has focused on improving “average-case” fairness. On the other hand, fairness at the extreme ends of the spectrum, which often signifies lasting and impactful shifts in societal attitudes, has received significantly less emphasis. Leveraging the statistics of extreme value theory (EVT), we propose a novel fairness criterion called extreme counterfactual discrimination (ECD). This criterion estimates the worst-case amounts of disadvantage in outcomes for individuals solely based on their memberships in a protected group. Utilizing tools from search-based software engineering and generative AI, we present a randomized algorithm that samples a statistically significant set of points from the tail of ML outcome distributions even if the input dataset lacks a sufficient number of relevant samples. We conducted several experiments on four ML models (deep neural networks, logistic regression, and random forests) over 10 socially relevant tasks from the literature on algorithmic fairness. First, we evaluate the generative AI methods and find that they generate sufficient samples to infer valid EVT distribution in 95% of cases. Remarkably, we found that the prevalent bias mitigators reduce the average-case discrimination but increase the worst-case discrimination significantly in 35% of cases. We also observed that even the tail-aware mitigation algorithm-MiniMax-Fairness-increased the worst-case discrimination in 30% of cases. We propose a novel ECD-based mitigator that improves fairness in the tail in 90% of cases with no degradation of the average-case discrimination. We hope that the EVT framework serves as a robust tool for evaluating fairness in both average-case and worst-case discrimination.

While current machine-learning-based AI techniques have been spectacularly successful, their present applications still leave many important open questions – for example, how to make their results more reliable or, at least, how to gauge how reliable is each AI recommendation. In this paper, we argue that to fully answer these questions, we need to go beyond the current AI techniques, and that in this development, systems-, human-, and cybernetics-based ideas not only naturally appear, they seem to provide a way to the desired answers.

Elon Musk's successful “move fast and break things” strategy is based on the fact that in many cases, we do not need to satisfy all usual constraints to be successful. By sequentially trying smaller number of constraints, he finds the smallest number of constraints that are still needed to succeed - and using this smaller number of constrains leads to a much cheaper (and thus, more practical) design. In this strategy, Musk relies on his intuition - which, as all intuitions, sometimes works and sometimes doesn't. To replace this intuition, we propose an algorithm that minimizes the worst-case cost of finding the smallest number of constraints.

The fuzzy-possibilistic product partition c-means (FPPPCM) algorithm was proposed as a robust solution to the c-means clustering problem, in which outlier data behave similarly to distant objects in gravity systems. Although FPP-PCM reliably provides fine partitions when its parameters are well chosen, things can be difficult when it is not initialized properly. To avoid such cases, this paper proposes a self-tuning version of the FPPPCM algorithm, which incorporates some cluster size controlling variables into the objective function that allow for the adjustment of the so-called possibilistic penalty terms during the alternative optimization process. The proposed method was evaluated using four standard test datasets in three different scenarios: (1) no added noise; (2) a single outlier added; (3) multiple noisy items added. The partitions provided by the proposed algorithm were evaluated based on cluster purity, normalized mutual information and adjusted Rand index, and was compared with the outcome of previous clustering models. The proposed method performed better or at least at the same quality level at previous ones, while reducing the number of parameters the user is responsible for.

We present a deep learning model for finding human-understandable connections between input features. Our approach uses a parameterized, differentiable activation function, based on the theoretical background of nilpotent fuzzy logic and multi-criteria decision-making (MCDM). The learnable parameter has a semantic meaning indicating the level of compensation between input features. The neural network determines the parameters using gradient descent to find human-understandable relationships between input features. We demonstrate the utility and effectiveness of the model by successfully applying it to classification problems from the UCI Machine Learning Repository.

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 several (A) alpha-cuts of the input. However, a straightforward application of this idea requires A times longer computation time than each range estimation – and for complex data processing algorithms, each range computation is already time-consuming. In this paper, we show how to compute all the desired ranges much faster.

In many cases, experts are much more accurate when they estimate the ratio of two quantities than when they estimate the actual values. For example, it is difficult to accurately estimate the height of a person on a photo, but if we have two people standing side by side, we can easily estimate to what extent one of them is taller than the other one. To get accurate estimates, it is therefore desirable to use such ratio estimates. Empirical analysis shows that to obtain the most accurate results, we need to compare all the objects with either the "best" object – i.e., the object with the largest value of the corresponding quantity – or the "worst" object – i.e., the object with the smallest value of this quantity. In this paper, we provide a theoretical explanation for this empirical observation.

At first glance, the larger the object, the larger should be its effect on the surroundings – in particular, the larger should be its effect on the surrounding flow. However, in many practical situations, we observe the opposite effect: smaller-size particles affect the flow much more than larger-size particles. This seemingly counterintuitive phenomena has been observed in many situations: lava flow in the volcanoes, air circulation in tornadoes, blood flow in a body, the effect of fish on water circulation in the ocean, and the effect of added particles on seeping water that damages historic buildings. In this paper, we show that all these phenomena can be explained in natural geometric terms.

In many econometric situations, we can predict future values of relevant quantities by using an empirical formula known as exponential Almon lag. While this formula is empirically successful, there have been no convincing theoretical explanation for this success. In this paper, we provide such a theoretical explanation based on general invariance ideas.

In many applications of intelligent computing, we need to choose an appropriate function – e.g., an appropriate re-scaling function, or an appropriate aggregation function. In applications of intelligent techniques, the problem of selecting an optimal function is usually too complex or too imprecise to be solved analytically, so the best functions are found empirically, by trying a large number of alternatives. In this paper, we show that in many such cases, the resulting empirical choice can be explained by natural invariance ideas. Example range from applications to building blocks of intelligent techniques – such as aggregation (including hierarchical aggregation) and averaging – to method-specific (polynomial fuzzy approach, pooling and averaging in deep learning) and domain-specific application, such as describing relative position of 2D and 3D objects, gauging segmentation quality, and perception of delay in public transportation.

Entropy is a natural measure of randomness. It progresses from its smallest possible value 0 – when we have a deterministic case in which one alternative i occurs with probability 1 (pi= 1), to the largest possible value which is attained at a uniform distribution p1= … = pn= 1/n. Intuitively, both in the deterministic case and in the uniform distribution case, there is not much variety in the distribution, while in the intermediate cases, when we have several different values pi, there is a strong variety. Entropy does not seem to capture this notion of variety. In this paper, we discuss how we can describe this intuitive notion.

Most of our decisions are based on the notion of similarity: we use a decision that helped in similar situations. From this viewpoint, it is important to have, for each pair of situations or objects, a numerical value describing similarity between them. This is called a similarity measure. In some cases, the only information that we can use to estimate the similarity value is some natural distance measure d(a,b). In many such situations, empirical data shows that the similarity measure 1/(1+d) is very effective. In this paper, we provide two explanations for this effectiveness.

In the late 1990s, researchers analyzed what distinguishes great companies from simply good ones. They found several features that are typical for great companies. Interestingly, most of these features seem counter-intuitive. In this paper, we show, on the qualitative level, that from the algorithmic viewpoint, many of these features make perfect sense. Some of the resulting explanations are simple and straightforward, other explanations rely on complex not-well-publicized results from theoretical computer science.

Elementary formulas for propagating information about means and variances through mathematical expressions have long been used by analysts. Yet the precise implications of such information are rarely articulated. This paper explores distribution-free techniques for risk analysis that do not require simulation, sampling or approximation of any kind. We describe best-possible bounds on risks that can be inferred given only information about the range, mean and variance of a random variable. These bounds generalise the classical Chebyshev inequality in an obvious way. We also collect in convenient tables several formulas for propagating range and moment information through calculations involving 7 binary convolutions (addition, subtraction, multiplication, division, powers, minimum, and maximum) and 9 unary transformations (scalar multiplication, scalar translation, exponentiation, natural and common logarithms, reciprocal, square, square root and absolute value) commonly encountered in risk expressions. These formulas are rigorous rather than approximate, and in most cases are either exact or mathematically best-possible. The formulas can be used effectively even when only interval estimates of the moments are available. Although most discussions of moment propagation assume stochastic independence among variables, this paper shows the assumption to be unnecessary and generalises formulas for the case when no assumptions are made about dependence, and when correlations are partially known. Along with partial means and variances, we show how interval covariances may be propagated and tracked through expressions.

Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA…

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 to a regression problem more accurate.

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 - and, moreover, we show that it is optimal in some reasonable sense.

Most multi-layer neural networks used in deep learning utilize rectified linear neurons. In our previous papers, we showed that if we want to use the exact same activation function for all the neurons, then the rectified linear function is indeed a reasonable choice. However, preliminary analysis shows that for some applications, it is more advantageous to use different activation functions for different neurons - i.e., select a family of activation functions instead, and select the parameters of activation functions of different neurons during training. Specifically, this was shown for a special family of squashing functions that contain rectified linear neurons as a particular case. In this paper, we explain the empirical success of squashing functions by showing that the formulas describing this family follow from natural symmetry requirements.

Often, we use original expert estimates to compute estimates of related quantities. In many practical situations, it is desirable to know how accurate is the resulting estimate. There are many techniques for computing this accuracy: we can use simple probabilistic ideas and we can use simple fuzzy ideas. Strangely enough, these two reasonable techniques lead to drastically different results. Which of them is correct? Our practical tests show that none of these two methods is perfect: probabilistic approach usually underestimates uncertainty, while the fuzzy approach overestimates it. This looks similar to many cases that motivated Zadeh to promote the idea of soft computing – a combination of different uncertainty techniques. To get a more adequate combination technique, we analyzed the general problem of combining accuracy estimates and came up with a 1-parametric family of techniques that contains probabilistic and fuzzy as particular cases – and that indeed works better on several practical examples that each of the original two techniques.

Fuzzy logic is normally used to describe the uncertainty of human knowledge and human reasoning. Physical phenomena are usually described by probabilistic models. In this paper, we show that in extremal conditions, when the concentrations are very large, some formulas describing physical interactions become fuzzy-type. We also show the observable consequences of such fuzzy-type formulas: they lead to bursts of gravitational waves.

Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

To process huge amounts of data, one possibility is to combine some data points into granules, and then process the resulting granules. For each group of data points, if we try to include all data points into a granule, the resulting granule often becomes too wide and thus rather useless; on the other case, if the granule is too narrow, it includes only a few of the corresponding point – and is, thus, also rather useless. The need for the trade-off between coverage and specificity is formalized as the principle of justified granularity. The specific form of this principle depends on the selection of a measure of specificity. Empirical analysis has show that exponential and power law measures of specificity are the most adequate. In this paper, we show that natural symmetries explain this empirically observed efficiency.

If all we know about normalized fuzzy sets is which set is a subset of which, will we be able to detect crisp sets? It is known that we can do it if we allow all possible fuzzy sets, including non-normalized ones. In this paper, we show that a similar detection is possible if we only allow normalized fuzzy sets. We also show that we can detect type-1 fuzzy sets based on the subsethood ordering of normalized interval-valued fuzzy sets.

Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

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 possible, moreover, that it is possible under three different definitions of order between numerical values.

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 the corresponding process is discontinuous. In this paper, we show that by applying the soft computing techniques to translate this imprecise knowledge into a precise strategy, we can get an efficient algorithm for detecting discontinuities; its efficiency is shown on the example of detecting a fault based on the seismic signals.

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 big difference on the result. Surprisingly, it turns out that in some practical situations, we get a much more accurate estimates if we, vice versa, take into account the bounds - and ignore all the information about the probabilities. In this paper, we explain the corresponding algorithms. and we show, on a practical example, that using this algorithm can indeed lead to a drastic improvement in estimation accuracy.

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 corresponding predictions if we take into account chaotic and delayed character of the corresponding system.

Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

Possibilistic logic is a well-known framework for dealing with uncertainty and reasoning under inconsistent or prioritized knowledge bases. This paper deals with conditioning uncertain information where the weights associated with formulas are in the form of sets of uncertainty degrees. The first part of the paper studies set-valued possibility theory where we provide a characterization of set-valued possibilistic logic bases and set-valued possibility distributions by means of the concepts of compatible possibilistic logic bases and compatible possibility distributions respectively. The second part of the paper addresses conditioning set-valued possibility distributions. We first propose a set of three natural postulates for conditioning set-valued possibility distributions. We then show that any set-valued conditioning satisfying these three postulates is necessarily based on conditioning the set of compatible standard possibility distributions. The last part of the paper shows how one can efficiently compute set-valued conditioning over possibilistic knowledge bases.

One of the main techniques used to de-noise and deblur signals and images is regularization, which is based on the fact that signals and images are usually smoother than noise. Traditional Tikhonov regularization assumes that signals and images are differentiable, but, as Mandelbrot has shown in his fractal theory, many signals and images are not differentiable. To de-noise and de-blur such images, researchers have designed a heuristic method of ℓp-regularization. ℓp-regularization leads to good results, but it is not used as widely as should be, because it lacks a convincing theoretical explanation - and thus, practitioners are often reluctant to use it, especially in critical situations. In this paper, we show that fuzzy techniques provide a theoretical explanation for the ℓp-regularization. Fuzzy techniques also enables us to come up with natural next approximations to be used when the accuracy of the ℓp-based denoising and de-blurring is not sufficient.

The focus of this paper is to clarify the concepts of solutions of linear equations in interval, probabilistic, and fuzzy sets setting for real world tasks. There is a fundamental difference between formal definitions of the solutions and physically meaningful concepts of solution in applied tasks, when equations have uncertain components. For instance, a formal definition of the solution in terms of Moore interval analysis can be completely irrelevant for solving a real world task. We show that formal definitions must follow a meaningful concept of the solution in the real world. The paper proposed several formalized definitions of the concept of solution for the linear equations with uncertain components in the interval, probability and fuzzy set terms that can be interpreted in the real world tasks.

In some cases, a membership function μ(x) represents an unknown number, but in many other cases, it represents an unknown crisp set. In this case, for each crisp set S, we can estimate the degree μ(S) to which this set S is the desired one. A natural question is: once we know the values μ(S) corresponding to all possible crisp sets S, can we reconstruct the original membership function? In this paper, we show that the original membership function μ(x) can indeed be uniquely reconstructed from the values μ(S).

In many real-life situations, we need to reconstruct a blurred image in situations when no information about the blurring is available. This problem is known as the problem of blind deconvolution. There exist techniques for solving this problem, but these techniques are not rotation-invariant. Thus, the result of using this technique may change with rotation. So, if we rotate the image a little bit, the method, in general, leads to a different deconvolution result. Therefore, even when the original reconstruction is optimal, the reconstruction of a rotated image will be different and, thus, not optimal. To improve the quality of image decomposition, it is desirable to modify the current state-of-the art techniques by making them rotation-invariant. In this paper, we show how this can be done, and we show that this indeed improves the quality of blind deconvolution.

In situations when several participants collaborate with each other, it is desirable to come up with a fair way to divide the resulting gain between the participants. Such a fair way was proposed by John von Neumann and Oscar Morgenstern, fathers of the modern game theory. However, in some situations, the von Neumann-Morgenstern solution does not exist. To cover such situations, we propose to use a fuzzy-inspired hierarchical version of the von Neumann-Morgenstern (NM) solution. We prove that, in contrast to the original NM solution, the hierarchical version always exists.

Fuzzy techniques are a successful way to handle expert knowledge, enabling us to capture different degrees of experts' certainty in their statements. To use fuzzy techniques, we need to describe experts' degrees of certainty in numerical terms. Some experts can provide such numbers, but others can only describe their degrees by using natural-language words like “very”, “somewhat”, “to some extent”, etc. In general, all we know about these word-valued degrees is that there is a natural partial order between these degrees: e.g., “very small” is clearly smaller than “somewhat small”. In this paper, we propose a natural way to transform such a partial order between degrees into numerical values.

Fuzzy techniques have been originally designed to describe imprecise (“fuzzy”) expert knowledge. Somewhat surprisingly, fuzzy techniques have also been successfully used in situations without expert knowledge, when all we have is data. In this paper, we explain this surprising phenomenon by showing that the need for optimal processing of data (including crisp data) naturally leads to fuzzy and neural data processing techniques. This result shows the potential of fuzzy data processing. To maximally utilize this potential, we need to provide an operational meaning of the corresponding fuzzy degrees. We show that such a meaning can be extracted from the above justification of fuzzy techniques. It turns out that, in contrast to probabilistic uncertainty, the natural operational meaning of fuzzy degrees is indirect—similarly to the operational meaning of geometry and physics in general relativity.

To more adequately gauge the student's knowledge, it is desirable to take into account not only whether the student's answers on the test are correct or nor, but also how confident the students are in their answers. For example, a situation when a student gives a wrong answer, but understands his/her lack of knowledge on this topic, is not as harmful as the situation when the student is absolutely confident in his/her wrong answer. In this paper, we use the general decision making theory to describe the best way to take into account the student's degree of certainty when evaluating the test results.

One of the main objectives of ecology is to analyze, maintain, and enhance the bio-diversity of different ecosystems. To be able to do that, we need to gauge bio-diversity. Several semi-heuristic diversity indices have been shown to be in good accordance with the intuitive notion of bio-diversity. In this paper, we provide a theoretical justification for these empirically successful techniques. Specifically, we show that the most widely used techniques - Simpson index - can be justified by using simple fuzzy rules, while a more elaborate justification explains all empirically successful diversity indices.

To describe expert uncertainty, it is often useful to go beyond additive probability measures and use non-additive (fuzzy) measures. One of the most widely used classes of such measures is the class of Sugeno λ-measures. Their success is somewhat paradoxical, since from the purely mathematical viewpoint, these measures are - in some reasonable sense - equivalent to probability measures. In this paper, we explain this success by showing that while (1) mathematically, it is possible to reduce Sugeno measures to probability measures, but (2) from the computational viewpoint, using Sugeno measures is much more efficient. We also show that among all fuzzy measures which are equivalent to probability measures, Sugeno measures (and a slightly more general family of measures) are the only ones with this efficiency property.

To gauge the risk corresponding to a possible disaster, it is important to know both the probability of this disaster and the expected damage caused by such potential disaster (“expected shortfall”). Both these measures of risk are easy to estimate in the ideal case, when we know the exact probabilities of different disaster strengths. In practice, however, we usually only have a partial information about these probabilities: we may have an interval (or, more generally, fuzzy) uncertainty about these probabilities. In this paper, we show how to efficiently estimate the expected shortfall under such interval and/or fuzzy uncertainty.

In some practical situations – e.g., when treating a new illness – we do not have enough data to make valid statistical conclusions. In such situations, it is necessary to use expert knowledge – and thus, it is beneficial to use fuzzy techniques that were specifically designed to process such knowledge. At first glance, it may seem that in situations when we have large amounts of data, the relative importance of expert knowledge should decrease. However, somewhat surprisingly, it turns out that expert knowledge is still very useful in the current age of big data. In this paper, we explain how exactly (and why) expert knowledge is useful, and we overview efficient methods for processing this knowledge. This overview is illustrated by examples from environmental science, geosciences, engineering (in particular, aircraft maintenance and underwater robots), and medicine.

In the traditional fuzzy logic, the experts' degrees of confidence in their statements is described by numbers from the interval [0,1]. These degree have a clear intuitive meaning. Somewhat surprisingly, in some applications, it turns out to be useful to also consider different numerical degrees - e.g., complex-valued degrees. While these complex-valued degrees are helpful in solving practical problems, their intuitive meaning is not clear. In this paper, we provide a possible explanation for the success of complex-valued degrees which makes their use more intuitively understandable - namely, we show that these degrees naturally appear due to the approximate nature of the traditional fuzzy methodology.

In this paper, we explore one of the possible ways to make decisions under uncertainty: namely, we explain how to define a fair price for a participation in such a decision, and then select an alternative for which the corresponding fair price is the largest. This idea is explained on the examples of interval uncertainty, set-valued, fuzzy, and Z-number uncertainty.

In the traditional fuzzy logic, the expert's degree of confidence d(A&B) in a complex statement A&B (or A V B) is uniquely determined by his/her degrees of confidence d(A) and d(B) in the statements A and B, as f&(d(A), d(B)) for an appropriate "and"-operation (t-norm). In practice, for the same degrees d(A) and d(B), we may have different degrees d(A&B) depending on the relation between A and B. The best way to take this relation into account is to explicitly elicit the corresponding degrees d(A&B) and d(A V B), i.e., to come up with a "double" fuzzy set. If we only elicit information about pairs of statements, then we still need to estimate, e.g., the degree d(A & B & C) based on the known values d(A), d(B), d(C), d(A & B), d(A & C), and d(B & C). In this paper, we explain how to produce such "and"-operations for "double" fuzzy sets — and how to produce similar "or"-operations.

Traditional probabilistic description of uncertainty is based on additive probability measures. To describe non-probabilistic uncertainty, it is therefore reasonable to consider non-additive measures. An important class of non-additive measures are possibility measures, for which μ(A ∪ B) = max(μ(A), μ(B)). In this paper, we show that possibility measures are, in some sense, universal approximators: for every ε > 0, every non-additive measure which satisfies a certain reasonable boundedness property is equivalent to a measure which is ε-close to a possibility measure.

In this paper, we describe how to estimate relative spatial resolution of different maps or images of the same area under uncertainty. We consider probabilistic and fuzzy approaches and we show that both approaches lead to the same estimate - which makes us somewhat more confident that this joint result is reasonable.

In many practical situations, e.g., in signal processing, image processing, analysis of temporal data, it is very useful to use fuzzy (F-) transforms. In an F-transform, we first replace a function x(t) by a few local averages (this is called forward F-transform), and then reconstruct the original function from these averages (this is called inverse F-transform). While the formula for the forward F-transform makes perfect intuitive sense, the formula for the inverse F-transform seems, at first glance, somewhat counter-intuitive. On the other hand, its empirical success shows that this formula must have a good justification. In this paper, we provide such a justification - a justification which is based on formulating a reasonable compression-based criterion.

In fuzzy logic, there are two main approaches to eliciting membership degrees: an approach based on polling experts, and a Likert-scale approach, in which we ask experts to indicate their degree of certainty on a scale - e.g., on a scale form 0 to 10. Both approaches are reasonable, but they often lead to different membership degrees. In this paper, we analyze the relation between these two approaches, and we show that this relation can be made much clearer if we use models from quantum computing.

Computer systems based on fuzzy logic are capable of generating a reliable output even when handling inaccurate input data by applying a rule based system. The main contribution of this paper is to show that quantum computing can be used to extend the class of fuzzy sets. The central idea associates the states of a quantum register to membership functions (mFs) of fuzzy subsets, and the rules for the processes of fuzzyfication are performed by unitary quantum transformations. Thus, this paper describes multi-dimensional quantum registers, associated to mFs on the unitary interval U, in order to introduce a novel interpretation of aggregations found in fuzzy set theory. In particular, t-norms and t-conorms based on quantum gates allows the modeling and interpretation of union, intersection, difference and implication among fuzzy sets, also including an expression for the class of fuzzy S-implications. Furthermore, an interpretation of the symmetric sum was achieved by considering the quantum register sum operator.

An expert opinion describes his or her opinion about a quantity by using imprecise ("fuzzy") words from a natural language, such as "small", "medium", "large", etc. Each of these words provides a rather crude description of the corresponding quantity. A natural way to refine this description is to assign degrees to which the observed quantity fits each of the selected words. For example, an expert can say that the value is reasonable small, but to some extent it is medium. In this refined description, we represent each quantity by a tuple of the corresponding degrees. Once we have such a tuple-based information about several quantities x1, xm, and we know that another quantity y is related to x1 by a known relation y = f (x1,..., xm), it is desirable to come up with a resulting tuple-based description of y. In this paper, we describe why a seemingly natural idea for computing such a tuple does not work, and we show how to modify this idea so that it can be used.

In many practical situations, we encounter physical quantities like time for which there is no fixed starting point for measurements: physical properties do not change if we simply change (shift) the starting point. To describe knowledge about such properties, it is desirable to select membership functions which are similarly shift-invariant. We show that while we cannot require that each membership function is shift-invariant, we can require that the linear space of all linear combinations of given membership functions is shift-invariant. We describe all such shift-invariant families of membership functions, and we show that they are naturally related to the corresponding formulas of chemical kinetics.

In many applications, we know the function f (x1 xn), we know the intervals xi of possible values of each quantity xi, and we are interested in the range of possible values of y = f (x1 xn), this problem is known as the problem of interval computations. In other applications, we know the function f (x1 xn), we know the fuzzy sets Xi that describe what we know about each quantity xi, and we are interested in finding the fuzzy set Y corresponding to the quantity y = f (x1 xn), this problem is known as the problem of fuzzy computations. There are many efficient algorithms for solving these problems, however, most of these algorithms implicitly assume that each quantity xi can take any real value within its range. In practice, some quantities are discrete: e.g., xi can describe the number of people. In this paper, we provide feasible algorithms for interval, set, and fuzzy computations for such discrete inputs.

It is well known that an arbitrary continuous function on a bounded set - e.g., on an interval [a; b] - can be, with any given accuracy, approximated by a polynomial or by a piece-wise polynomial function (spline). Usually, polynomials are described as linear combinations of monomials. It turns out that in many computational problems, it is more efficient to represent each polynomial as a Bernstein polynomial - e.g., for functions of one variable, a linear combination of terms (x - a)k· (b - x)n-k. In this paper, we provide a simple fuzzy-based explanation of why Bernstein polynomials are often more efficient than linear combinations of monomials, and we show how this informal explanation can be transformed into a precise mathematical explanation.

To properly gauge the extent of poverty in a country or in a region, economists use semi-heuristic poverty measures such as the Foster-Greer-Thorbecke (FGT) metric. These measures are used because it was empirically shown that they capture the commonsense meaning of the extent of poverty better than previously proposed measures. However, without a theoretical justification, we cannot guarantee that these semi-heuristic measures will work in other situations as well. So, it is desirable to look for poverty measures which can be theoretically justified. In this paper, we first use fuzzy techniques to provide a commonsense interpretation of FGT poverty measures, and then show that how this informal interpretation can be transformed into a formal justification of the FGT property measures - from certain reasonable assumptions.

To enhance learning, it is desirable to also let students learn from each other, e.g., by working in groups. It is known that such groupwork can improve learning, but the effect strongly depends on how we divide students into groups. In this paper, based on a first approximation model of student interaction, we describe how to optimally divide students into groups so as to optimize the resulting learning. We hope that, by taking into account other aspects of student interaction, it will be possible to transform our solution into truly optimal practical recommendations.

Often, about the same real-life system, we have both measurement-related probabilistic information expressed by a probability measure P(S) and expert-related possibilistic information expressed by a possibility measure M(S). To get the most adequate idea about the system, we must combine these two pieces of information. For this combination, R. Yager—borrowing an idea from fuzzy logic—proposed to use a t-norm f&(a,b) such as the product f&(a,b)=a· b, i.e., to consider a set function f(S)=f&(P(S),M(S)). A natural question is: can we uniquely reconstruct the two parts of knowledge from this function f(S)? In our previous paper, we showed that such a unique reconstruction is possible for the product t-norm; in this paper, we extend this result to a general class of t-norms. © 2011 Wiley Periodicals, Inc.

Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

International Journal of Uncertainty, Fuzziness and Knowledge-Based SystemsVol. 20, No. 06, pp. 943-944 (2012) No AccessInterval Methods in Knowledge Representation♢Vladik KreinovichVladik KreinovichDepartment of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USAhttps://doi.org/10.1142/S0218488512970070Cited by:0 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail ♢ Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA. Remember to check out the Most Cited Articles! Check out our titles on Fuzzy Logic & Z-Numbers With a wide range of areas, you're bound to find something you like. FiguresReferencesRelatedDetails Recommended Vol. 20, No. 06 Metrics History PDF download

Best Fit Least-Squares (BFLS) is a required technique for many STEM subjects. It is a method to compute a linear model for a set of data points. Due to its utility, BFSL is frequently taught to STEM students before they have sufficient mathematical experience to follow the mechanics of a derivation. Not only does this fail to produce procedural and conceptual understandings, but also it encourages students to view formulae and algorithms as things to be looked up, rather than derived. This is discourages students from developing productive dispositions. In this paper, we describe the “Close Fit Rod-Spring” (CFRS) approach to the problem. This approach computes the resting position of a rigid rod, which is connected by vertically oriented springs to a set of data points. This method of derivation results in two linear equations that may be solved for the slope and intercept of the best fit line. The result is equivalent to BFSL. However, it is achieved in an intuitively understandable and mathematically accessible way for high school students.

Measurements are never absolutely accurate, the measurement result x̃ is, in general, different from the actual (unknown) values x of the corresponding quantity. In many practical problems, we only know upper bounds Δ on the measurement errors equation. In such situations, once we know the measurement result, the only conclusion that we can make about the actual value x is that this value belongs to the interval [x̃ - Δ, x̃ + Δ]. There exist many efficient algorithms for processing such interval data. However, these algorithms usually assume that all the measurement results are valid. In reality, due to factors such as sensor malfunction, some measurement results may be way off (outliers), for which the difference between x̃ and x is much larger than the upper bound Δ on the measurement error. In this paper, we overview the algorithmic problems related to processing interval sensor data in the presence of outliers. Our case study - for which we develop and analyze these algorithms - is localization of underwater robots, a problem in which a significant number of measurement results are outliers.

Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

In many practical situations, we need to optimize several objectives under the positivity constraints. For example, in meteorological and environmental studies, it is important to collect various types of data, such as temperature and wind speed and direction, from weather stations. For maintenance purposes, it is convenient to place instruments that collect different weather data on the same weather station. Thus, we need to find the “best” location for a weather station. The “best” means, for example, that the external influences, such as flux of cars passing on nearby road, have a minimal impact on the measurement results. There are several such criteria, so we face a multi-objective optimization problem. In this paper, we show that traditional approaches for solving such problems - such as the weighted sum approach - are not fully adequate for solving our problem. We show that fuzzy heuristics lead to a more adequate approach - of using a generalized form of Nash bargaining solution. We then prove that under reasonable assumptions of scale-invariance, the generalized Nash bargaining solution is the only adequate solution for the general problem of multi-objective optimization under positivity constraints - and, in particular, for the problem of selecting an optimal location for a weather station.

In the 1920s, Pavel Urysohn proved his famous lemma (sometimes referred to as “first non-trivial result of point set topology”). This lemma was instrumental in proving that under reasonable conditions, every topological space can be metrized. Motivated by the success of pseudo-metric spaces in General Relativity, Urysohn started working on an extension of his lemma and of the metrization theorem to (causality-)ordered topological spaces and corresponding pseudo-metrics. By the 1970s, general space-time versions of Urysohn's lemma and metrization theorem have been proven. However, the proofs of these results are not natural - they looks like clever tricks, not like a direct consequence of the definitions. Since one of the main objectives of this activity is to come up with useful applications to physics, we desire more natural versions of these proofs. In this paper, we show that fuzzy logic leads to such natural proofs.

In some cases, our uncertainty about a quantity can be described by an interval of its possible values. If we have two or more pieces of interval information about the same quantity, then we can conclude that the actual value belongs to the intersection of these intervals. In general, we may need a fuzzy number to represent our partial knowledge. A fuzzy number can be viewed as a collection of intervals (α-cuts) corresponding to different degrees α ∈[0,1]. In practice, we can only store finitely many α-cuts. Usually, we only store the lower and upper α-cuts (corresponding to α = 0 and α = 1) and use linear interpolation - i.e., use trapezoidal fuzzy numbers. However, the intersection of two trapezoidal fuzzy numbers is, in general, not trapezoidal. One possible approach is to simply take an intersection of lower and alpha α-cuts, but this approach underestimates the resulting membership function. In this paper, we propose a more accurate approach that uses the Least Squares Method to provide a better linear approximation to the resulting membership function. While this method provides a more accurate trapezoidal description of the intersection, it has its own drawbacks: e.g., this approximation method makes the corresponding “knowledge fusion” operation non-associative. We prove, however, that this “drawback” is inevitable: specifically, we prove that a perfect solution is not possible, and that any improved trapezoidal approximation to intersection (fusion) of trapezoidal fuzzy numbers leads to non-associativity.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

In 1951, K.J. Arrow proved that, under certain assumptions, it is impossible to have group decision-making rules that satisfy reasonable conditions like symmetry. This Impossibility Theorem is often cited as a proof that reasonable group decision-making is impossible. We start our article by remarking that Arrow's result covers only those situations when the only information we have about individual preferences is their binary preferences between the alternatives. If we follow the main ideas of modern decision making and game theory and also collect information about the preferences between lotteries (i.e., collect the utility values of different alternatives), then reasonable decision-making rules are possible, e.g., Nash's rule in which we select an alternative for which the product of utilities is the largest possible. We also deal with two related issues: how we can detect individual preferences if all we have is preferences of a subgroup and how we take into account the mutual attraction between participants. © 2008 Wiley Periodicals, Inc.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: V. Kreinovich, Department of Computer Science, University of Texas, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: V. Kreinovich, Department of Computer Science, University of Texas, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

In the ideal case of complete knowledge, for each property Pi(such as “high fever”, “headache”, etc.), we know the exact set Siof all the objects that satisfy this property. In practice, we usually only have partial knowledge. In this case, we only know the set Siof all the objects about which we know that Piholds and the set Siabout which we know that Pimay hold (i.e., equivalently, that we have not yet excluded the possibility of Pi). This pair of sets is called a set interval. Based on the knowledge of the original properties, we would like to describe the set S of all the values that satisfy some combination of the original properties: e.g., high fever and headache and not rash. In the ideal case when we know the exact set Siof all the objects satisfying each property, it is sufficient to apply the corresponding set operation (composition of union, intersection, and complement) to the known sets Si. In this paper, we describe how to compute the class S of all possible sets S.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: V. Kreinovich, Department of Computer Science, University of Texas, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: V. Kreinovich, Department of Computer Science, University of Texas, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: V. Kreinovich, Department of Computer Science, University of Texas, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: V. Kreinovich, Department of Computer Science, University of Texas, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: V. Kreinovich, Department of Computer Science, University of Texas, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: V. Kreinovich, Department of Computer Science, University of Texas, El Paso, TX 79968, USA.

The paper investigates the feasibility of fuzzy models application in measurement procedures. It considers the problem of measurement information fusion from different sources, when one of the sources provides predictions regarding approximate values of the measured variables or their combinations. Typically, this information is given by an expert but may be mined from available data also. This information is formalized as fuzzy prediction models and is used in combination with the measurement results to improve the measurement accuracy. The properties of the modified estimates are studied in comparison with the conventional ones. The conditions when fuzzy models application can achieve a significant accuracy gain are derived, the gain value is evaluated, and the recommendations on fuzzy prediction model production and formalization in practical applications are given.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: V. Kreinovich, Department of Computer Science, University of Texas, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: V. Kreinovich, Department of Computer Science, University of Texas, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: V. Kreinovich, Department of Computer Science, University of Texas, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: V. Kreinovich, Department of Computer Science, University of Texas, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: V. Kreinovich, Department of Computer Science, University of Texas, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: V. Kreinovich, Department of Computer Science, University of Texas, El Paso, TX 79968, USA.

Existing approaches to timing analysis under uncertainty are based on restrictive assumptions. Statistical STA techniques assume that the full probabilistic distribution of parameter uncertainty is available; in reality, the complete probabilistic description often cannot be obtained. In this paper, a new paradigm for parameter uncertainty description is proposed as a way to consistently and rigorously handle partially available descriptions of parameter uncertainty. The paradigm is based on a theory of interval probabilistic models that permit handling uncertainty that is described in a distribution-free mode- just via the range, the mean, and the variance. This permits effectively handling multiple real-life challenges, including imprecise and limited information about the distributions of process parameters, parameters coming from different populations, and the sources of uncertainty that are too difficult to handle via full probabilistic measures (e.g. on-chip supply voltage variation). Specifically, analytical techniques for bounding the distributions of probabilistic interval variables are proposed. Besides, a provably correct strategy for fast Monte Carlo simulation based on probabilistic interval variables is introduced. A path-based timing algorithm implementing the novel modeling paradigm, as well as handling the traditional variability descriptions, has been developed. The results indicate the proposed algorithm can improve the upper bound of the 90 th-percentile circuit delay, on average, by 5.3 % across the ISCAS’85 benchmark circuits, compared to the worst-case timing estimates that use only the interval information of the partially specified parameters.

It has been observed that when people make crude estimates, they feel comfortable choosing between alternatives that differ by a half-order of magnitude (e.g., were there 100, 300, or 1000 people in the crowd?) and less comfortable making a choice on a more detailed scale, with finer granules, or on a coarser scale (like 100 or 1000). In this article, we describe two models of choosing granularity in commonsense estimates, and we show that for both models, in the optimal granularity, the next estimate is three to four times larger than the previous one. Thus, these two optimization results explain the commonsense granularity. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 843–855, 2006.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: V. Kreinovich, Department of Computer Science, University of Texas, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: V. Kreinovich, Department of Computer Science, University of Texas, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: V. Kreinovich, Department of Computer Science, University of Texas, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: V. Kreinovich, Department of Computer Science, University of Texas, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: V. Kreinovich, Department of Computer Science, University of Texas, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: V. Kreinovich, Department of Computer Science, University of Texas, El Paso, TX 79968, USA.

According to the traditional probability theory, events with a positive but very small probability can occur (although very rarely). For example, from the purely mathematical viewpoint, it is possible that the thermal motion of all the molecules in a coffee cup goes in the same direction, so this cup will start lifting up. In contrast, physicists believe that events with extremely small probability cannot occur. In this paper, we show that to get a consistent formalization of this belief, we need, in addition to the original probability measure, to also consider a maxitive (possibility) measure

The age of fossil species in samples recovered from a well that penetrates an undisturbed sequence of sedimentary rocks increases with depth. The results of biostratigraphic analysis of such a sequence consist of several age-depth values - both known with interval (or fuzzy) uncertainty - and we would like to find, for each possible depth, the interval of the possible values of the corresponding age. A similar problem of bounding an intervally (fuzzily) defined function under monotonicity constraint occurs in many other application areas. In this paper, we provide an efficient algorithm for solving this problem

In a 1944 book that started game theory (and the mathematical approach to conflict resolution), von Neumann and Morgenstern proposed the notion of a solution. When the situation changes, the old solution is often no longer a solution, so it needs to be updated. In practical applications, it is usually desirable to keep the solution change “minimal” in some reasonable sense. We show that for a seemingly straightforward formalization of this minimality, checking whether a change is minimal is NP-hard. We also show that by representing the notion of a solution as a collection of revision rules, we can produce a reasonable notion of minimality for which there exists a feasible algorithm for checking the minimality of the update. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 939–956, 2005.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: VLADIK KREINOVICH, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: VLADIK KREINOVICH, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

The paper considers the problem of improving accuracy and reliability of measurement information acquired by sensor networks. It offers the way of integrating sensor measurement results with association information available or a priori derived at aggregating nodes. The models applied for describing both sensor results and association information are reviewed with consideration given to both neuro-fuzzy and probabilistic models and methods. The information sources, typically available in sensor systems, are classified according to the model (fuzzy or probabilistic), which seems more feasible to be applied. The integration problem is formalized as an optimization problem.

A 1990 article in the American Mathematical Monthly has shown that most combinatorial identities of the type described in Monthly problems can be solved by known identity checking algorithms. A natural question arises: are these algorithms always feasible or can the number of computational steps be so big that application of these algorithms sometimes is not physically feasible? We prove that the problem of checking identities is nondeterministic polynomial (NP) hard, and thus (unless NP = P) for every algorithm that solves it, there are cases in which this algorithm would require exponentially long running time and thus will not be feasible. This means that no matter how successful computers are in checking identities, human mathematicians will always be needed to check some of them. © 2004 Wiley Periodicals, Inc.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: VLADIK KREINOVICH, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

To design data processing algorithms with the smallest average processing time, we need to know what this "average" stands for. At first glance, it may seem that real-life data are really "chaotic", and no probabilities are possible at all: today, we may apply our software package to elementary particles, tomorrow - to distances between the stars, etc. However, contrary to this intuitive feeling, there are stable probabilities in real-life data. This fact was first discovered in 1881 by Simon Newcomb who noticed that the first pages of logarithm tables (that contain numbers starting with 1) are more used than the last ones (that contain numbers starting with 9). To check why, he took all physical constants from a reference book, and counted how many of them start with 1. An intuitive expectation is that all 9 digits should be equally probable. In reality, instead of 11 %, about 30% of these constants turned out to be starting with 1. In general, the fraction or constants that start with a digit d can be described as ln(d + 1) - ln(d). We describe a new interval computations-related explanation for this empirical fact, and we explain its relationship with lifetime of the Universe and with the general problem of determining subjective (fuzzy) probabilities on finite and infinite intervals.

The paper considers the problem of measurement information fusion from different sources, when one of the sources is an information about approximate values of the measured variables or their combinations. The information is given with fuzzy models and is used in combination with the measurement results. The properties of the modified estimates are studied in comparison with the conventional ones. The conditions when an expert's information application can give a high gain are derived, the gain value is estimated, the recommendations to an expert on making predictions are given. The possible gain in measurement result efficiency in geophysical applications is analyzed.

In fuzzy logic, every word or phrase describing uncertainty is represented by a real number from the interval [0, 1]. There are only denumerable many words and phrases and continuum many real numbers; thus, not every real number corresponds to some common sense degree of uncertainty. In this article, for several fuzzy logics, we describe which numbers are describing such degrees, i.e., in mathematical terms, which real numbers are definable in the corresponding fuzzy logic. © 2003 Wiley Periodicals, Inc.

This section is maintained by Vladik Kreinovich. Please send your abstract (or copies of papers that you want to see reviewed here), to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send you abstract (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send you abstract (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send you abstract (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send you abstract (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

Uncertainty is very important in risk analysis. A natural way to describe this uncertainty is to describe a set of possible values of each unknown quantity (this set is usually an interval), plus any additional information that we may have about the probability of different values within this set. Traditional statistical techniques deal with the situations in which we have a complete information about the probabilities; in real life, however, we often have only partial information about them. We therefore need to describe methods of handling such partial information in risk analysis. Several such techniques have been presented, often on a heuristic basis. The main goal of the paper is to provide a justification for a general second-order formalism for handling different types of uncertainty.

Logical inference starts with concluding that if B implies A, and B is true, then A is true as well. To describe probabilistic inference rules, we must therefore define the probability of an implication "A if B". There exist two different approaches to defining this probability, and these approaches lead to different probabilistic inference rules: we may interpret the probability of an implication as the conditional probability P(A|B), in which case we get Bayesian inference. We may also interpret this probability as the probability of the material implication A /spl or/ /spl not/ B in which case we get different inference rules. We develop a general approach to describing the probability of an implication, and we describe the corresponding general formulas, of which Bayesian and material implications are particular cases. This general approach is naturally formulated in terms of t-norms, a term which is normally encountered in fuzzy logic.

For large aerospace structures, it is extremely important to detect faults, and nondestructive testing is the only practical way to do it. Based on measurements of ultrasonic waves, Eddy currents, magnetic resonance, etc., we reconstruct the locations of the faults. The best (most efficient) known statistical methods for fault reconstruction are not perfect. We show that the use of expert knowledge-based granulation improves the quality of fault reconstruction.

A natural way to test the structural integrity of a pavement is to send signals with different frequencies through the pavement and compare the results with the signals passing through an ideal pavement. For this comparison, we must determine how, for the corresponding mixture, the elasticity E depends on the frequency f in the range from 0.1 to 10/sup 5/ Hz. It is very expensive to perform measurements in the high frequency area (above 20 Hz). To avoid these measurements, we can use the fact that for most of these mixtures, when we change a temperature, the new dependence changes simply by scaling. Thus, instead of performing expensive measurements for different frequencies, we can measure the dependence of E on moderate frequencies f for different temperatures, and then combine the resulting curves into a single "master" curve. In this paper, we show how fuzzy techniques can help to automate this "combination".

There are two main fuzzy system methodologies for translating expert rules into a logical formula: In Mamdani's methodology, we get a DNF formula (disjunction of conjunctions), and in a methodology which uses logical implications, we get, in effect, a CNF formula (conjunction of disjunctions). For both methodologies, universal approximation results have been proven which produce, for each approximated function f(x), two different approximating relations RDNF(x, y) and RCNF(x, y). Since, in fuzzy logic, there is a known relation FCNF(x) ≤ FDNF(x) between CNF and DNF forms of a propositional formula F, it is reasonable to expect that we would be able to prove the existence of approximations for which a similar relation RCNF(x, y) ≤ RDNF(x, y) holds. Such existence is proved in our paper. © 2002 Wiley Periodicals, Inc.

We propose a new way to generalize logical operations from the discrete classical logic to a continuous fuzzy logic, namely we propose to define derivatives for the discrete case, and then to use these derivatives to derive the continuous operations. We show that this natural approach leads to "algebraic" fuzzy operations a/spl middot/b and a+b-a/spl middot/b.

In a series of papers and a book, Margaliot and Langholz (1999, 2000) proposed a hyperbolic approach to fuzzy control, in which they apply a certain hyperbolic non-linear transformation to the original variables. We consider all possible nonlinear transformations of this type and show that this hyperbolic transformation is indeed optimal.

We describe a new symmetry-based derivation of centroid defuzzification for fuzzy logic and fuzzy control.

Many different "and"- and "or"-operations have been proposed for use in fuzzy logic. It is therefore important to select, for each particular application, the operations which are the best for this particular application. Several papers discuss the optimal choice of "and"- and "or"-operations for fuzzy control, when the main criterion is to get the stablest control (or the smoothest or the most robust or the fastest-to-compute). In reasoning applications, however, it is more appropriate to select operations which are the best in reflecting human reasoning, i.e., operations which are "the most logical". In this paper, we explain how we can use logic motivations to select fuzzy logic operations, and show the consequences of this choice. As one of the unexpected consequences, we get a surprising relation with the entropy techniques, well known in probabilistic approach to uncertainty.

How is fuzzy logic usually formalized? There are many seemingly reasonable requirements that a logic should satisfy: e.g., since A and B and B and A are the same, the corresponding AND-operation should be commutative. Similarly, since A and A means the same as A, we should expect that the AND-operation should also satisfy this property, etc. It turns out to be impossible to satisfy all these seemingly natural requirements, so usually, some requirements are picked as absolutely true (like commutativity or associativity), and others are ignored if they contradict the chosen ones. This idea leads to a neat mathematical theory, but the analysis of real-life expert reasoning shows that all the requirements are only approximately satisfied. We should require all of these requirements to be satisfied to some extent. We show the preliminary results of analyzing such operations. In particular, we show that non-associative operations explain the empirical 7/spl plusmn/2 law in psychology according to which a person can normally distinguish between no more than 7 plus or minus 2 classes.

In order to adequately process satellite and radar information, it is necessary to find the exact correspondence between different types of images and between these images and the existing maps. In other words, we need to reference these images. In this paper, we propose new methods for automatic referencing of satellite and radar images.

In reasoning about time and duration, researchers often use Allen's interval algebra. This algebra describes possible relations between 1-D intervals. An interval can precede the other one, follow the other one, start the other one, etc. This algebra describes the relationship between different intervals in terms of words from natural language. To give a natural language description of 2D images, it is desirable to develop a similar approach for describing the relationship between 2-D objects in a picture. In their recent papers, J. Keller and his collaborators proposed a new approach based on a simulation of a "force" between these objects. In this paper, we show that their force formula is theoretically optimal.

Geospatial databases often contain erroneous measurements. For some such databases such as gravity databases, the known methods of detecting erroneous measurements-based on regression analysis-do not work well. As a result, to clean such databases, experts use manual methods which are very time-consuming. In this paper, we propose a (natural) "localized" version of regression analysis as a technique for automatic cleaning. We illustrate the efficiency of this technique on the example of the gravity database.

Several universal approximation and universal representation results are known for non-Boolean multivalued logics such as fuzzy logics. In this paper, we show that similar results can be proven for multivalued Boolean logics as well. © 2001 John Wiley & Sons, Inc.

Chess is probably the best known example of a game which is “biased”—in the sense that whoever starts the game has an advantage. From the commonsense viewpoint, the resulting bias should be the same whether we allow the players to play as usual: 1st, 2nd, 1st, 2nd, or whether we allow each player to make two moves at the same time. However, in practice, if we allow each player to make two moves in succession, the bias increases. In this paper, we provide a theoretical explanation for this empirical phenomenon. © 2001 John Wiley & Sons, Inc.

Traditional fuzzy logic uses real numbers as truth values. This description is not always adequate, so in interval-valued fuzzy logic, we use pairs (t/sup -/, t/sup +/) of real numbers, t/sup -/ /spl les/ t/sup +/, to describe a truth value. To make this description even more adequate, instead of using real numbers to described each value t/sup -/ and t/sup +/, we can use intervals, and thus get fuzzy values which can be described by 4 real numbers each. We can iterate this procedure again and again. The question is: can we get an arbitrary partially ordered set in this manner or an arbitrary lattice? In this paper, we show that although we cannot thus generate arbitrary lattices, we can actually generate an arbitrary partially ordered set in this manner. In this sense, the "intervalization" operation is indeed universal.

Uncertainty of complex-valued physical quantities z=x+y can be described by complex fuzzy sets. Such sets can be described by membership functions /spl mu/(x, y) which map the universe of discourse (complex plane) into the interval [0, 1]. The problem with this description is that it is difficult to directly translate into words from natural language. To make this translation easier, several authors have proposed to use, instead of a single membership function for describing the complex number, several membership functions which describe different real-valued characteristics of this number, such as its real part, its imaginary part, its absolute value, etc. The quality of this new description strongly depends on the choice of these real-valued functions, so it is important to choose them optimally. We formulate the problem of optimal choice of these functions and show that, for all reasonable optimality criteria, the level sets of optimal functions are straight lines and circles. This theoretical result is in good accordance with our numerical experiments, according to which such functions indeed lead to a good description of complex fuzzy sets.

Fuzzy information processing systems start with expert knowledge which is usually formulated in terms of words from natural language. This knowledge is then usually reformulated in computer friendly terms of membership functions, and the system transforms these input membership functions into the membership functions which describe the result of fuzzy data processing. It is then desirable to translate this fuzzy information back from computer-friendly membership functions language to human-friendly natural language. In general, this is difficult even in a 1-D case, when we are interested in a single quantity y; however, the fuzzy research community has accumulated some expertise of describing the resulting 1-D membership functions by words from natural language. The problem becomes even more complicated in 2-D and multi-D cases, when we are interested in several quantities y/sub 1/,...,y/sub m/, because there are fewer words which describe the relation between several quantities than words describing a single quantity. To reduce this more complicated multi-D problem to a simpler (although still difficult) 1-D case, Zadeh proposed (1966) to use words to describe fuzzy information about different combinations y=f(y/sub 1/,...,y/sub m/) of the desired variables. This idea is similar to the use of marginal distributions in probability theory. The corresponding terms are called shadows of the original fuzzy set. The main question is: do we lose any information in this translation? Zadeh has shown that under certain conditions, the original fuzzy set can be uniquely reconstructed from its shadows. We prove that for appropriately chosen shadows, the reconstruction is always unique. Thus, if we manage to describe the original membership function by linguistic terms which describe different combinations y, this description is lossless.

Sparse rule base and interpolation have been proposed as possible solution to alleviate the geometric complexity problem of large fuzzy set. However, no formal method to extract sparse rule base is yet available. This paper combines the recently introduced Cartesian representation of membership functions and a mountain method-based clustering technique for the extraction. A case study is included to demonstrate the effectiveness of the approach.

One of the reasons why fuzzy methodology is successful is that fuzzy systems are universal approximators, i.e., we can approximate an arbitrary continuous function within any given accuracy by a fuzzy system. In some practical applications (e.g., in control), it is desirable to approximate not only the original function, but also its derivatives (so that, e.g., a fuzzy control approximating a smooth control will also be smooth). In our paper, we show that for any given accuracy, we can approximate an arbitrary smooth function by a fuzzy system so that not only the function is approximated within this accuracy, but its derivatives are approximated as well. In other words, we prove that fuzzy systems are universal approximators for smooth functions and their derivatives. ©2000 John Wiley & Sons, Inc.

Function approximation is a very important practical problem: in many practical applications, we know the exact form of the functional dependence y=f(x1,…,xn) between physical quantities, but this exact dependence is complicated, so we need a lot of computer space to store it, and a lot of time to process it, i.e., to predict y from the given xi. It is therefore necessary to find a simpler approximate expression g(x1,…,xn)≈f(x1,…,xn) for this same dependence. This problem has been analyzed in numerical mathematics for several centuries, and it is, therefore, one of the most thoroughly analyzed problems of applied mathematics. There are many results related to approximation by polynomials, trigonometric polynomials, splines of different type, etc. Since this problem has been analyzed for so long, no wonder that for many reasonable formulations of the optimality criteria, the corresponding problems of finding the optimal approximations have already been solved. Lately, however, new clustering-related techniques have been applied to solve this problem (by Yager, Filev, Chu, and others). At first glance, since for most traditional optimality criteria, optimal approximations are already known, the clustering approach can only lead to non-optimal approximations, i.e., approximations of inferior quality. We show, however, that there exist new reasonable criteria with respect to which clustering-based function approximation is indeed the optimal method of function approximation. © 2000 John Wiley & Sons, Inc.

It is often necessary to divide a certain amount of money between n participants, i.e., to assign, to each participant, a certain portionwi≥0 of the whole sum (so that w1+⋯+ wn=1). In some situations, from the fairness requirements, we can uniquely determine these "weights" wi. However, in some other situations, general considerations do not allow us to uniquely determine these weights, we only know the intervals[Formula: see text] of possible fair weights. We show that natural fairness requirements enable us to choose unique weights from these intervals; as a result, we present an algorithm for fair division under interval uncertainty.

In this paper, we formalize and prove the statement that coincidences cannot be accidental, a statement that underlies many useful heuristics in mathematics and physics. Our proof uses a version of Kolmogorov complexity, a technique originally developed to describe randomness and "accidentalness."

If we view fuzzy logic as a logic, i.e., as a particular case of a multi-valued logic, then one of the most natural questions to ask is whether the corresponding propositional logic is decidable, i.e., does there exist an algorithm that, given two propositional formulas F and G, decides whether these two formulas always have the same truth value. It is known that the simplest fuzzy logic, in which &=min and ∨=max, is decidable. In this paper, we prove a more general result: that all propositional fuzzy logics with algebraic operations are decidable. We also show that this result cannot be generalized further, e.g., no deciding algorithm is possible for logics in which operations are algebraic with constructive (nonalgebraic) coefficients. ©1999 John Wiley & Sons, Inc.14: 935–947, 1999

In many practical problems, we must optimize a set function, i.e., find a set A for which f(A)→max, where f is a function defined on the class of sets. Such problems appear in design, in image processing, in game theory, etc. Most optimization problems can be solved (or at least simplified) by using the fact that small deviations from an optimal solution can only decrease the value of the objective function; as a result, some derivative must be equal to 0. This approach has been successfully used, e.g., for set functions in which the desired set A is a shape, i.e., a smooth (or piece-wise smooth) surface. In some real-life problems, in particular, in the territorial division problem, the existing methods are not directly applicable. For such problems, we design a new simple differential formalism for optimizing set functions. ©1999 John Wiley & Sons, Inc.

Many applications of probability theory are based on the assumption that, as the number of cases increase, the relative frequency of cases with a certain property tends to a number – probability that this property is true. L. Zadeh has shown that in many real-life situations, the frequency oscillates and does not converge at all. It is very difficult to describe such situations by using methods from traditional probability theory. Fuzzy logic is not based on any convergence assumptions and therefore, provides a natural description of such situations. However, a natural next question arises: how can we describe this oscillating behavior? Since we cannot describe it by using a single parameter (such as probability), we need to use a multi-D formalism. In this paper, we describe an optimal formalism for describing such oscillations, and show that it complements traditional probability techniques in the same way as fractals complement smooth curves and surfaces.

This paper introduces a new computational model for SequenceL. The model is extended through the addition of a notion of the VRAM model. The computational model introduced together with the SequenceL language provides for a good expression of distributed behaviour.

One of the most important parts of designing an expert system is elicitation of the expert's knowledge. This knowledge usually consists of facts and rules. Eliciting these rules and facts is relatively easy: the more complicated task is assigning weights (numerical or interval-valued degrees of belief) to different statements from the knowledge base. Experts often cannot quantify their degrees of belief, but they can order them (by suggesting which statements are more reliable). It is, therefore, reasonable to try to reconstruct the degrees of belief from such an ordering.In this paper, we analyze when such a reconstruction is possible, whether it lead to unique values of degrees of belief, and how computationally complicated the corresponding reconstruction problem can be. © 1998 John Wiley & Sons, Inc.

In fuzzy logic, there are several methods of representing implication in terms of &, ∨, and ¬; in particular, explicit representations define a class of S implications, implicit representations define a class of R implications. Some reasonable implication operations have been proposed, such as Yager's ab, that are difficult to represent as S or R implications. For such operations, a new class of representations has recently been proposed, called A implications, for which the relationship between implications and the basic operations &, ∨, and ¬ is even more complicated. A natural question is: Is this complexity really necessary? In other words, is it true that A operations cannot be described as S or R operations, or they can, but we simply have not found these representations? In this paper we show that yes, the complexity is necessary, because there are operations that cannot be represented in a simpler form. © 1998 John Wiley & Sons, Inc.

In science and engineering, there are “paradoxical” cases in which we have some arguments in favor of some statement A (so the degree to which A is known to be true is positive (nonzero)), and we have some arguments in favor of its negation ¬A, and we do not have enough information to tell which of these two statements is correct. Traditional fuzzy logic, in which “truth values” are described by numbers from the interval [0, 1], easily describes such “paradoxical” situations: the degree a to which the statement A is true and the degree 1−a to which its negation ¬A is true can both be positive. In this case, if we use traditional fuzzy &-operations (min or product), the “truth value” a&(1−a) of the statement A&¬A is positive, indicating that there is some degree of inconsistency in the initial beliefs. When we try to use fuzzy logic to formalize expert reasoning in the humanities, we encounter the problem that is humanities, in addition to the above-described paradoxical situations caused by the incompleteness of our knowledge, there are also true paradoxes, i.e., statements that are perceived as true and false at the same time. For such statements, A&¬A=“true.” The corresponding equality a&(1−a)=1 is impossible in traditional fuzzy logic (where a&(1−a) is always≤0.5), so, to formalize such true paradoxes, we must extend the set of truth values from the interval [0, 1]. In this paper we show that such an extension can be achieved if we allow truth values to be complex numbers. © 1998 John Wiley & Sons, Inc.

This section is maintained by Vladik Kreinovich. Please send your abstract (or copies of papers that you want to see reviewed here), to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstract (or copies of papers that you want to see reviewed here), to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstract (or copies of papers that you want to see reviewed here), to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstract (or copies of papers that you want to see reviewed here), to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstract (or copies of papers that you want to see reviewed here), to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

Textbooks often claim that quantum mechanics explained the periodic system: namely, the actual configuration of electronic orbits that is responsible for the element's chemical properties can be described as the one that minimizes the total energy, and the energy of each configuration can be computed by using quantum mechanics. However, a careful analysis of this explanation reveals that, in addition to the basic equations of quantum mechanics, we need some heuristic rules that do not directly follow from quantum physics. One reason why additional heuristics are necessary is that the corresponding numerical equations are extremely difficult to solve, and as we move to atoms with larger and larger atomic numbers Z, they become even more difficult. Moreover, as Z grows, we must take relativistic effects into consideration, and this means going from partial differential equations to even more mathematically difficult operator equations. In this paper, we show that if instead of the (often impossible) numerical optimization, we consider the (available) ordinal information, we can then explain the observed periodic system.

We humans usually think in words; to represent our opinion about, e.g., the size of an object, it is sufficient to pick one of the few (say, five) words used to describe size (“tiny,” “small,” “medium,” etc.). Indicating which of 5 words we have chosen takes 3 bits. However, in the modern computer representations of uncertainty, real numbers are used to represent this “fuzziness.” A real number takes 10 times more memory to store, and therefore, processing a real number takes 10 times longer than it should. Therefore, for the computers to reach the ability of a human brain, Zadeh proposed to represent and process uncertainty in the computer by storing and processing the very words that humans use, without translating them into real numbers (he called this idea granularity). If we try to define operations with words, we run into the following problem: e.g., if we define “tiny” + “tiny” as “tiny,” then we will have to make a counter-intuitive conclusion that the sum of any number of tiny objects is also tiny. If we define “tiny” + “tiny” as “small,” we may be overestimating the size. To overcome this problem, we suggest to use nondeterministic (probabilistic) operations with words. For example, in the above case, “tiny” + “tiny” is, with some probability, equal to “tiny,” and with some other probability, equal to “small.” We also analyze the advantages and disadvantages of this approach: The main advantage is that we now have granularity and we can thus speed up processing uncertainty. The main disadvantage is that in some cases, when defining symmetric associative operations for the set of words, we must give up either symmetry, or associativity. Luckily, this necessity is not always happening: in some cases, we can define symmetric associative operations. © 1997 John Wiley & Sons, Inc.

Please send your abstracts or copies of papers that you want to see reviewed here to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts or copies of papers that you want to see reviewed here to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

Usually, expert systems use numbers to describe the experts' degree of belief in their statements. In practice, however, it is difficult to assign an exact numerical value to the expert's degree of belief. At best, we can get an interval of possible values. This fact leads to the use of interval-valued degree of belief. When intervals are used to describe degrees of belief, then computations with intervals must be used to process them. In this paper, we describe applications of such interval computations to expert systems and to intelligent control.

This paper presents a rationale for providing hardware supported functions of more than two variables for processing incomplete knowledge and fuzzy knowledge. The result is in contrast to Kolmogorov's (1957) theorem in the numerical (nonfuzzy) case.

The formal concept of logical equivalence in fuzzy logic, while theoretically sound, seems impractical. The misinterpretation of this concept has led to some pessimistic conclusions. Motivated by practical interpretation of truth values for fuzzy propositions, we take the class (lattice) of all subintervals of the unit interval [0, 1] as the truth value space for fuzzy logic, subsuming the traditional class of numerical truth values from [0, 1]. The associated concept of logical equivalence is stronger than the traditional one. Technically, we are dealing with much smaller set of pairs of equivalent formulas, so that we are able to check equivalence algorithmically. The checking is done by showing that our strong equivalence notion coincides with the equivalence in logic programming. © 1996 John Wiley & Sons, Inc.

With this issue, we start a new section: short abstract of papers and books on interval methods in knowledge representation. The experience of several conferences on interval computations and fuzzy systems, including the 1995 International Workshop on Applications of Interval Computations (February 1995, El Paso, TX, USA), has shown that there are many areas of knowledge representation where interval methods are applied, and many interesting results of these applications, areas and results that are often not widely known to the knowledge representation community. These papers are published in different journals and conference proceedings, and it is difficult to trace them all. In view of this difficulty, we decided to provide the readers of IJUFKS with short abstract of these papers (something like an ongoing annotated bibliography). We strongly believe that the information about the current applications of interval methods is of interest to this community. For the reasons expressed above, we are currently, more probably, not covering all relevant papers. To increase the coverage, we need your help. If you know of any recent papers devoted to the applications of interval methods to knowledge representation, please send reference to Vladik Kreinovich at [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA. If you have writtend your own reviews of such papers, or if you would like to write such reviews, please contact Vladik as well. Authors, please send information and/or copies of your own applications papers (papers in French, Russian, and German are also welcome). Abstracts should ideally in LATEX or TEX, but ASCII is also acceptable This is a new section, and we want the readers' input about how to make it better. Any suggestions and recommendations will be highly welcome.

In this issues, we continue to publish abstracts and reviews of recents papers on interval methods in knowledge representation. In knowledge representation, intervals are used for two main purposes: • to describe durations of events; and • to describe uncertainty of measurement results and expert estimates of different quantities; often, we do not know the exact value of a quantity, but we know its lower and upper bounds (e.g., we may not know the exact value of someone's weight, but we may know that this weight is in between 140 and 160 pounds). An important case of this uncertainty occurs in knowledge elicitation, when we ask experts to numerically estimate their degrees of belief in their own statements; in this case, it is often difficult for an expert to estimate this degree of belief precisely, but an expert can often provide us with an interval of possible values. The reviews are collected by Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA, email [email protected]

This section is maintained by Vladik Kreinovich. Please send your abstract or copies of papers that you want to see reviewed here to [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.

This paper addresses mathematical aspects of fuzzy logic. The main results obtained in this paper are: 1. the introduction of a concept of normal form in fuzzy logic using hedges; 2. using Kolmogorov’s theorem, we prove that all logical operations in fuzzy logic have normal forms; 3. for min-max operators, we obtain an approximation result similar to the universal approximation property of neural networks.

Fuzzy control is a methodology that translates “if”-“then” rules, Aji (x1) &…& Ajn(xn) → Bj(u), formulated in terms of a natural language, into an actual control strategy u(x). Implication of uncertain statements is much more difficult to understand than “and,” “or,” and “not.” So, the fuzzy control methodologies usually start with translating “if”-“then” rules into statements that contain only “and,” “not,” and “or.” the first such translation was proposed by Mamdani in his pioneer article on fuzzy control. According to this article, a fuzzy control is reasonable iff one of the rules is applicable, i.e., either the first rule is applicable (A11(x1) &…& A1n(xn) & B1(u)), or the second one is applicable, etc. This approach turned out to be very successful, and it is still used in the majority of fuzzy control applications. However, as R. Yager noticed, in some cases, this approach is not ideal: Namely, if for some x, we know what u(x) should be, and add this crisp rule to our rules, then the resulting fuzzy control for this x may be different from the desired value u(x). to overcome this drawback, Yager proposed to assign priorities to the rules, so that crisp rules get the highest priority, and use these priorities while translating the rules into a control strategy u(x). In this article, we show that a natural modification of Mamdani's approach can solve this problem without adding any ad hoc priorities. © 1995 John Wiley & Sons, Inc.

The notion of a transitive closure of a fuzzy relation is very useful for clustering in pattern recognition, for fuzzy databases, etc. It is based on translating the standard definition of transitivity and transitive closure into fuzzy terms. This definition works fine, but to some extent it does not fully capture our understanding of transitivity. the reason is that this definition is based on fuzzifying only the positive side of transitivity: if R(a, b) and R(b, c), then R(a, c); but transitivity also includes a negative side: if R(a, b) and not R(a, c), then not R(b, c). In classical logic, this negative statement follows from the standard “positive” definition of transitivity. In fuzzy logic, this negative part of the transitivity has to be formulated as an additional demand. In the present article, we define a strongly transitive fuzzy relation as the one that satisfies both the positive and the negative parts of the transitivity demands, prove the existence of strong transitive closure, and find the relationship between strongly transitive similarity and clustering. © 1995 John Wiley & Sons, Inc.

The original idea of fuzzy logic and other soft computing methodologies was to handle the situations in which our knowledge is not precise. Usually, real numbers are used to describe degrees of belief. In practice, only approximate values of the degrees of belief are known, while the existing soft computing formalisms are usually based on the assumption that we know the exact values of these degrees. This difference creates a gap between the theory and applications. In this paper, we outline the theoretical foundations aimed at bridging this gap.

Experts usually express their uncertainty by words of natural languages (like "perhaps", "for sure", etc). In the majority of expert systems and intelligent control systems, uncertainty of experts' statements is represented by a number from the interval [0, 1]. There are many different procedures that translate the words that experts use into numbers from [0, 1]. For one and the same word, different procedures can lead to different numbers. Some &– and ∨–operations are very sensitive to this difference in the sense that small changes in t(A) and t(B) can lead to absolutely different estimates for t(A&B) and t(A ∨ B). In view of that, it is reasonable to restrict ourselves to the operations that are the least sensitive to such changes. In this paper, we prove that ab and a + b − ab are "in the average" the least sensitive &– and ∨–operations. This result is in good accordance with the experimental data according to which in. many cases, these operations provide the best description of how experts actually think. We also show how this idea can be applied to other logical connectives (e.g., "not"), and to the choice of membership functions.

A mathematical explanation as to why fuzzy control is smoother and more stable than control by experts whose experience was used to design the fuzzy control is presented. The analysis indicates that fuzzy control is always continuous and exhibits better performance, even in complicated control situations.>