International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems Vol. 13, No. 2 (2005) 241-242 © World Scientific Publishing Company

INTERVAL METHODS IN KNOWLEDGE REPRESENTATION (Abstracts of recent papers)

This section is maintained by Vladik Kreinovich. Please send your abstracts (or copies of papers that you want to see reviewed here) to vladik@cs.utep.edu, or by regular mail to:

VLADIK KREINOVICH

Department of Computer Science University of Texas at El Paso

El Paso, TX 79968, USA.

D. Bamber, I. R. Goodman, and H. T. Nguyen, "Robust reasoning with rules that have exceptions: from second-order probability to augmentation via upper proba­bility, upper possibility, and directed graphs", Annals of Mathematics and Artificial Intelligence, 2005 (to appear).

In mathematical logic, a rule is something that is always true. In real life appli­cations in engineering and science, often, a rule is something for which exceptions are rare. As a result, not all the logical operations that we can do in mathematical logic can be applied to the practical rules. For example, in mathematical logic, we can have arbitrarily long deduction sequences - and the results will still be correct. If we know that A\ is true, that A\ implies A<i, that A<i implies As, etc., then we can conclude that A100 is true as well. In contrast, in real life, if we have only 99% certainty in each rule A\ —> A2, Ai —> As, etc., then AIOQ can be always false. This problem is important in the case of trust, when we trust our bank with a high probability, the bank trusts its clients, they trust their agents, etc., but if the chain of trust becomes too long, it is quite possible that one of the links in this chain of trust is fraudulent.

Usually, we only only the probability of rare exceptions to each rule, and we often have no information about the correlation between the validity of different rules. As a result, for a long deduction chain, we cannot compute the probability of the resulting rule or statement, we can only produce an interval of possible values of this probability. The authors show how we can compute the corresponding lower and upper probabilities. In particular, they show that in many practical situations, when we combine two rules, the upper probability of the exception to the resulting rule is equal to the maximum of the probabilities corresponding to individual rules -in other words, that the corresponding upper probabilities behave like possibilities.

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242 V. Kreinovich

0 . Castillo and P. Melin, "A new approach for plant monitoring using type-2 fuzzy logic and fractal theory", International Journal of General Systems, 2004, Vol. 33, No. 4, pp. 305-319.

In many application areas such as manufacturing, it is important to monitor the process, and if necessary, to diagnose the situation based on the results of this monitoring. One of the problems with monitoring is that it generates a lot of data, and the excess of data prevents us from efficiently using statistical, neural, and other efficient learning and data classification algorithms.

In some cases, an important criterion is the degree of smoothness of the data: e.g., if the behavior is normally smooth, then the appearance of random-like fluc­tuation may indicate some problems; vice versa, if harmless random fluctuations evolve into a smooth trend that threatens to take the system outside its stability area, it is not a good sign. One characteristic of the system's degree of smoothness is its fractal dimension. Crudely speaking, for each real number e, we count the smallest number of points N(e) such that any point from the analyzed curve is e-close to one of these points. Often, when e become small, we have N(e) ~ C • £~a ; this number a is called the fractal dimension of the curve.

For a straight line segment of length L, this smallest number is attained when we place points at the distances £, 3e, 5e, etc. from one of its ends. In this case, N(e) ~ L/{2e) = (L/2) • £ _ 1 , so the fractal dimension is 1. Similarly, a smooth curve has a fractal dimension 1. On the other hand, Brownian motion has a fractal dimension 1.5, and other random processes have fractal dimensions ranging from 1 to 2.

By asking experts to evaluate different curves, we can come up with fuzzy rules that relate the fractal dimensions of different monitored parameters with different possible diagnoses. One problem with this approach is that experts normally de­scribe their rules in terms of words from natural language like "small", "large", etc. We must therefore ask the experts to translate these words into computer-understandable numbers. When experts assign fuzzy values (i.e., values from the interval [0,1]) to the words, they produce somewhat different numbers for the same word: e.g., the same expert may assign different values to a statement "1.2 is a small deviation". Different numbers, in their turn, may lead to different diagnoses, so the diagnostics based on these numbers is not as reliable as we would like.

It turns out that while it is difficult for an expert to select an exact number describing his or her degree of confidence, experts are much better in describing the interval of possible numbers that describe their degrees of belief. In other words, the resulting second order (interval-valued) fuzzy logic provides a much more stable description of the expert opinions. Not surprisingly, second order fuzzy logic leads to a much better representation of expert rules - in particular, in plant monitoring via fractals.

For example, for the control of an electro-chemical process of battery manufac­turing, the use of fractals leads to a 10% improvement in efficiency, and the use of interval-valued fuzzy logic leads to another 2% improvement.

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