Fuzzy differences

Figure 8.25 Example of a fuzzy difference of two numbers, that is, about 20 minus about 14 gives about 6.

Fuzzy sets and fuzzy logic. Fuzzy sets differ from the normal crisp sets in the fact that their elements have partial membership (represented by a value between 0 an 1) in the set. Fuzzy logic differs from the binary logic by the fact that the truth values are represented by fuzzy sets. 

An overview over different applications of fuzzy set theory and fuzzy logic is given in [15] (see also Chapter IX, Section 1.5 in the Handbook). 

Using these operators, fuzzy inference mechanisms are then developed to manipulate rules that include fuzzy values. The largest difference between fuzzy inference and ordinary inference is that fuzzy inference aUows partial match of input and produces an interpolated output. This technology is useml in control also. See Ref. 94. 

An important aspect of fuzzy logic is the ability to relate sets with different universes of discourse. Consider the relationship... 

The next few steps are very similar to those required in any software project. One of the first stages is the clear definition of the knowledge domain. It must be clear which problems the expert system must solve. It is at this stage not the intention to define how this can be done. Clarity and specificity must be the major guides here. Fuzziness at this stage will, more than in classical software projects, have to be paid for later when different interpretations cause misunderstandings. Equally important is the clear definition of the end user(s). An expert system set up as decision support tool for professionals is totally different from an expert system that can be used as a training support for less professional people. 

It may be useful to point out a few topics that go beyond a first course in control. With certain processes, we cannot take data continuously, but rather in certain selected slow intervals (c.f. titration in freshmen chemistry). These are called sampled-data systems. With computers, the analysis evolves into a new area of its own—discrete-time or digital control systems. Here, differential equations and Laplace transform do not work anymore. The mathematical techniques to handle discrete-time systems are difference equations and z-transform. Furthermore, there are multivariable and state space control, which we will encounter a brief introduction. Beyond the introductory level are optimal control, nonlinear control, adaptive control, stochastic control, and fuzzy logic control. Do not lose the perspective that control is an immense field. Classical control appears insignificant, but we have to start some where and onward we crawl. 

Membership and probability have some points of contact, but they are not the same. In the probabilistic view, an object must be completely in one set or another and the probability defines the "chance" that the object will be a member of one particular class. If there is a 70 percent chance that a person is tall, he is more likely than not to be in the "tall" set, but if he is not in that set, he must be in a different one, such as "medium" or "very tall." In the fuzzy view, an object may span two or more sets, so the person would be "mainly" in the tall set and at the same time partly in the "medium" or the "very tall" set. 

The most important difference between the rules of fuzzy logic and those of probability becomes apparent when we consider the membership of an object in two or more classes simultaneously. Let us suppose that the fuzzy statement ... 

It may happen that only a single rule provides information about a particular output variable. When this is true, that rule can be used immediately as a measure of the membership for the variable in the corresponding set. In the enzyme problem, only one rule predicts that the rate is high, therefore, we can provisionally assign a membership of 0.2 for the rate in this fuzzy class. Often though, several rules provide fuzzy information about the same variable and these different outputs must be combined in some way. This is done by aggregating the outputs of all rules for each output variable. 

When we have evaluated all the rules, an output variable might belong to two or more fuzzy subsets to different degrees. For example, in the enzyme problem one rule might conclude that the rate is low to a degree of 0.2 and another that the rate is low to a degree of 0.8. In aggregation, all the fuzzy values that have been calculated for each output variable are combined to provide a consensus value for the membership of the output variable in each... 

Evaluation in the microprocessor may be carried out conventionally, in accordance with fuzzy logic algorithms or even on the basis of so called neural networks. To what extent such terms can be advertised to the end user as a type of quality criterion remains to be seen. For consumers these differences are largely of no consequence as they always receive clean, hygienic, problem-free laundry for which only the absolutely essential quantities of the required resources of water, energy and chemicals have been used. 

This energy difference may be interpreted in terms of two elliptical trajectories separated by Ae and with a phase lag between the leading and following edges of the element, ApAq that moves along the fuzzy trajectory. The two edges remain separated in time by a fixed amount A and define the elements A and B at -q0 and p respectively. 

Abstraction is the most useful technique a developer can apply being able to state the important aspects of a problem uncluttered by less-important detail.6 It s equally important to be able to trace how the more-detailed picture relates to the abstraction. We ve already seen some of the main abstraction techniques in Catalysis—the ability to treat a complex system as one object and to treat complex interactions as one action and yet state the outcome precisely. This approach contrasts with more-traditional design techniques in which abstract also tends to mean fuzzy, so you can t see whether a statement is right or wrong because it might have many different interpretations. 

Part of your job as an analyst is to reconcile different views when they conflict and to clarify concepts when they are fuzzy. Suppose your chent says, Several customers may be renting the same video. Customers can reserve videos, and when a reserved video is returned, we hold it off the shelves for the reserver to come and collect it. Somewhere in there are two uses of video one meaning an individual copy, and a second one meaning a title. Your job is to identify the two types. 

Roald Hoffmann, personal correspondence, letter of October 16, 1990. Also, R. Hoffmann, "Nearly Circular Reasoning," American Scientist, 76 (1988) 182185. And see Kurt Mislow and Paul Bickart, "An Epistemological Note on Chirality," Israel Journal of Chemistry 15 (197677) 16 "Thus chiral and achiral are used with two different connotations When the terms are applied to a geometric model, they are sharply defined, whereas when used in conjunction with observables, they necessarily entail a certain fuzziness" (6). 

DR. MARSHALL NEWTON (Brookhaven National Laboratory) I d like to ask a question about Hopfield s numbers. The alpha parameter from his 1974 paper [Hopfield, J. J. Proc. Natl. Acad. Sci., USA 1974, 7 1, 3640] was based not on the direct metal-metal interaction but rather was based on carbon-carbon overlap because it was two carbons which were closest together in his electron transport system. In contrast, Dr. Sutin gave some different numbers based on metal orbitals. Depending on whether one is interested in carbon-carbon overlap between two organic rings, or in direct metal-metal overlap, one might or might not opt for the Hopfield parameters. However, at the level of fuzziness which we have, it may not make any difference, I realize. 

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