Whatever happened to the fuzzy logic

Much of his time is spent solving problems or making decisions. He thus affects the environment and wants to change it in his mind.

Our decisions are often accompanied by uncertainties: On the one hand, the initial situation may be so complex that it cannot be precisely described. Second, you may not know exactly what you really want, and third, things you don't know, especially future events, could play a role. Our colloquial language reflects these uncertainties.

If we describe a problem in our normal language, we use it to create a verbal model of the situation, which reduces its complexity to a manageable human level. The three sources of uncertainty mentioned do not prevent us from making decisions - in contrast to the classic problem-solving approach by computer, where precision is crucial.

To clarify, imagine that you want to help someone reverse parking. You will then give instructions such as “a little to the left”, “a little further back”, “turn harder”, the driver will understand this and be able to react appropriately. Instructions of the form “turn the steering wheel 1 degree, 2 minutes, 1 second to the right”, “and now 13.5 millimeters to the rear”, “and now 14 degrees to the right” would neither be appropriate nor easy to carry out.

That precision does not make sense beyond a certain degree, the founder of fuzzy logic, the mathematician Lotfi A. Zadeh of the University of California at Berkeley, stated in 1973 in his “Principle of incompatibility”: “To the same extent, in As the complexity of a system increases, our ability to make precise and at the same time significant statements about its behavior decreases. Above a certain threshold, precision and significance (relevance) become almost mutually exclusive properties. "

If we want to use powerful tools to solve problems, nowadays these are mostly mathematical methods that are used on electronic data processing systems. To do this, the verbal model has to be translated into a formal one, i.e. into a language that can be understood by IT systems. As a rule, the situation is represented by a set of numbers and mathematical relationships between them (numerical model); Sometimes the numbers are replaced by more abstract quantities, which can be linked, for example, according to the rules of algebra (symbolic models). In this translation process, information is very often lost because you have to commit yourself: While a prerequisite may be more or less applicable or a goal may be more or less desirable in the colloquial description of the problem, precision is required in a formal description. Mathematical models are also usually bivalent; that is, they only differentiate between admissibility and inadmissibility, zero or one (dichotomous language). The problem transformed in this way can now be solved with mathematical methods, but under certain circumstances it does not correspond to the original situation.

If uncertainties are also to be taken into account, this is usually done using probability theory. Although it helps to model random uncertainties in an appropriate way, it is not intended for the fuzziness that is contained in our language. In addition, a linguistic model is content-determined, that is, both the words used and their meaning are essential. Formal models, on the other hand, are usually so abstract that the meaning of the model sizes no longer plays a role.

Uncertainty modeling

Considerable paradigm shifts have occurred in the last two to three decades, particularly in the treatment of uncertainty. While until about 20 years ago this was usually understood to mean the uncertainty caused by chance, various sources are now being discussed.

The sentence “The probability of hitting the target is 0.8” is obviously a probabilistic statement that concerns this random uncertainty and can also model it appropriately. But first, the event about which something is said (here: to hit the target) is to be formulated sharply and in two-valued terms; secondly, the probability itself has to be indicated numerically by a sharp number (in this case 0.8000). These two properties limit the applicability of probabilistic calculi considerably.

On the other hand, the meaning of expressions such as “big man”, “hot day”, “stable currency” is also uncertain, but not because of chance, but because of the linguistic (lexical) fuzziness. The meaning of these words results from the context, i.e. from the person of the speaker and the reference in which the respective expression is used.

In terms of a stable currency in South America, one understands something completely different than when a Central European currency is meant. "It is likely that we will have good success" is first of all a probabilistic statement. However, their random uncertainty is still superimposed on the linguistic of the fuzzy terms “good success” and “likely”.

The theory of fuzzy sets

In 1964, Zadeh wanted to fly from New York to San Francisco. At Kennedy Airport, he asked whether his plane was on time and received the answer: "Rest assured, your plane will not be too late." Zadeh politely sat down and tried to read the class from the departure board the "not too delayed aircraft" to define. The difficulties he had with this made him think longer about the problem, and a year later he wrote the first scientific paper on the theory of "fuzzy sets", the fuzzy sets. There are now around 15,000 publications in this field. The theory can be understood as a generalization of both classical set theory and classical two-valued logic. Let us first choose the set-theoretic approach:

In contrast to classical set theory, in which every object either belongs to a certain set or not, the individual elements of a fuzzy set can also belong to this to a certain degree, which is usually chosen in the interval between 0 and 1. The degree 1 means full membership and a degree of 0 means no membership. Between these extreme values ​​there is a continuous transition from “being an element” to “not being an element”.

This definition is used for the mathematical representation of vaguely delimited terms. If the term “comfortable room temperature” is to be described or - which amounts to the same thing - the amount of comfortable room temperatures is to be defined, this amount could only be a sharply defined interval in the classical theory, for example from 19 to 24 degrees Celsius. However, a temperature of 18.9 degrees, for example, would then be classified as not pleasant, which does not correspond to human perception in this form. In our example, 18.9 degrees would be assessed as “maybe not so pleasant anymore”, which means that this value, with a degree of membership of 0.8, could belong to the vague set of pleasant room temperatures. A fuzzy set is completely described by the membership function, which - in our example - indicates the extent to which it is perceived as pleasant for each temperature (Fig. 2).

The so-called linguistic variables are essential elements of fuzzy logic. Their values ​​are not numbers - as is the case with the usual numerical variables - but words and expressions (terms) of a natural language. Since words are not as precise as numbers, they are represented by fuzzy sets. An example of a linguistic variable is the room temperature mentioned above. This could take on the values ​​of a number of terms such as “cool”, “pleasant” or “warm”. Another example of a linguistic variable is the concept of the age of people (Fig. 2).

A set theory includes not only its objects, namely the sets, but also the links between these objects. In classical set theory, these are the operations intersection, union and complement. Your generalizations in the theory of fuzzy sets are to be defined via the membership functions of the sets to be manipulated. In 1965, for example, Zadeh proposed to determine the membership of the intersection of two fuzzy sets by using the minimum of the membership functions of the sets to be intersected element by element. Accordingly, the maximum was proposed for the association and the respective addition of the membership function to the value 1 for the complement.

As mentioned, the theory of fuzzy sets can be understood both as a generalized set theory and as a generalized dual logic. If the latter approach is chosen, one arrives at so-called fuzzy logic, for approximate and plausible reasoning. These are three stages, building on one another, in which an attempt is made, starting from our classical logic, to arrive at procedures which correspond much more to human inference. The bridge between set theory and logic is easy to establish by making one aware that the set theoretical average corresponds to the logical AND, the set theoretical union corresponds to the logical (non-exclusive) OR and the set theoretical complement corresponds to the logical negation.

The operator problem

If the theory of fuzzy sets is an extension of the well-known set theory, the usual connection rules for sets (commutative law, associative law, distributive law) must also apply to the extension to fuzzy. In 1972 the Americans Richard Bellmann and M. Giertz proved that this is the case for the minimum operator. For a somewhat more general formulation of the logical AND, Horst Hamacher demonstrated in 1976 that the mathematical model can only be a family of product operators (the Hamacher operators).

In the 1970s, however, people became aware of the fact that the “and” we use in our natural language, like all our words, is very much subject to linguistic or lexical fuzziness, particularly depending on the context. Accordingly, the mathematical models to be used in each case are also different and different from those which are to be used for the logical AND. Which model is appropriate in a specific situation is not a question of mathematics, but of psychology, more precisely of psycholinguistics. Studies carried out in Germany in the early 1970s, as well as other work, showed that the averaging operators are most likely to reproduce the colloquial AND.


Apart from many purely mathematical applications and the direct use of the theory of fuzzy sets, for example in the social sciences, the methodological approach can essentially be divided into two directions: knowledge-based and algorithmic applications.

Knowledge-based applications are based on the following basic idea: If it is difficult or impossible to formulate a problem mathematically adequately, or if an efficient algorithm is not available for the mathematical model of a problem, then one tries to fall back on human experience-based knowledge and, in an electronic data processing system, the human one To simulate problem-solving behavior. To do this, knowledge must be recorded and appropriately processed.

A number of methods have been developed for this, particularly in the field of artificial intelligence. The most popular representation works with systems of rules. They usually take the form: "IF condition, THEN action." They can be statements of the type "IF it rains, THEN the roads are wet" or statements of the form "IF the car does not start, THEN check the gasoline level". Rules of this kind are stored in the knowledge base; The human reasoning behavior is simulated in the so-called inference machine, which, based on its knowledge and observations, creates a diagnosis, a decision proposal or an instruction. This is the idea of ​​an expert system that has also been known in Germany since the early 1980s (Fig. 3).

The task of control engineering is to control technical processes with the help of observations (measurements). The results of these observations are usually in the form of numbers (values ​​for pressures, temperatures, and so on); the control signals must also be made available in numerical form. If one wants to use the idea of ​​the expert system here, the input data must first be converted into linguistic form (fuzzification) and, after processing by the expert system, the resulting linguistic statements (in the form of membership functions of linguistic variables) converted into a numerical form ( Defuzzification).

Algorithmic applications usually assume that there is a mathematical model or method for a problem. Since most of these methods work with sharply defined values ​​and many problems are only insufficiently described as a result, one tries to better adapt the models or methods to the problems with the help of the theory of fuzzy sets. Examples of this, which often go back to the 1970s, can be found in optimization (for example fuzzy linear programming), in pattern recognition (fuzzy clustering), in network technology and in other areas. This also includes fuzzy Petri nets, which are successfully used for process control or maintenance problems (compare the article by Hans-Peter Lipp on page 101).

Fuzzy control

The control technology application of fuzzy logic is implemented in Japan today in chemical production plants, high-rise elevators, waste incineration plants, industrial robots and underground trains as well as in products for private use: washing machines, video cameras, microwave ovens and anti-lock systems. These control systems are mostly based on a knowledge-based approach; algorithmic models are not available or not practicable, so that the use of knowledge-based systems leads to better or cheaper controls. Most industrial applications contain very small knowledge bases. They are therefore characterized by relatively simple structures.

A modern example of a very complex application is our fuzzy car (Fig. 1), which can independently avoid obstacles. In this model, the knowledge base contains some 100 rules. The entire sensor system consists of three very simple ultrasonic sensors that measure the distance to the front and to the right and left. If you realize that all the rules here have to be processed over and over again at short intervals in order to steer the car around unexpected obstacles, it becomes clear that fuzzy control can also be used for complex control tasks.

Data analysis

In the case of fuzzy data or information analysis, data structures are classified that can be very heterogeneous, such as handwritten characters, vectors, parameters, process history records, spectrograms or profiles. It is required wherever the input information is incomplete and the classes into which the structures are to be divided are only vaguely known or not known at all (compare the article by Steffen F. Bocklisch and Norman Bitterlich on page 99).

As a rule, people have to decide in such situations. In the case of fuzzy information analysis, one first defines certain patterns in advance; in the case of use, the system determines the extent to which a given situation is similar to these patterns and draws conclusions from this. Both qualitative and quantitative analyzes of such data structures, functional relationships or non-dichotomous image signals can be processed. In contrast to fuzzy control, both knowledge-based and algorithmic approaches are used, for example in the fuzzy cluster method or the fuzzy discriminant analysis.

Expert systems

While control and data analysis based on the principle of fuzzy logic are becoming increasingly widespread not only in Japan but also in Europe, the development of expert systems on this basis is only in its infancy. Since expert systems - in contrast to fuzzy control - do not have the option of confirming knowledge directly by observing a process, and since both input and output should take place in quasi-natural language, especially in expert-supporting systems, additional ones are required Problems that have not yet been fully resolved.This includes, above all, the development and selection of appropriate inference methods, the improvement of the linguistic approximation, the design of meaningful and powerful explanatory components and the improvement of the methods of knowledge acquisition, especially with regard to poorly structured knowledge.

At the Rheinisch-Westfälische Technische Hochschule Aachen, special systems for credit assessment, strategic planning and control of flexible manufacturing systems have already been developed; However, really powerful tools will only be on the market in one or two years. In addition, one has to wait and see whether a connection between fuzzy logic and neural networks will result in learning systems that can be used in practice.

Reluctant enforcement

Ten years after it was first published in 1965, the Fuzzy Set Theory was largely limited to the academic field. Only in England were rotary kilns in cement factories successfully controlled by fuzzy control as early as the first half of the 1970s. The first commercial fuzzy controller was offered by Smith in Denmark in 1976. As early as the second half of the 1970s, Fuzzy Control began to be used intensively in the field of control technology in Japan. In Europe, fuzzy optimization methods were still a rarity, although the first international magazine in the field of fuzzy sets was published in 1978.

In Japan there had been commercial applications of fuzzy control since the mid-1980s. In addition, the first software tools were probably created there, although they were not accessible to the rest of the world. In 1989 the first American software tool was offered. In the same year the LIFE Institute (Laboratory for International Fuzzy Engineering Research) was established in Japan, an amalgamation of 49 large Japanese companies in order to intensively advance applied research in this field.

It was not until 1990 that public interest aroused in Germany - primarily triggered by reports on Japanese products in the various media - (see my interview in Spektrum der Wissenschaft, March 1992, page 30). Numerous publications appeared; Symposia, seminars and meetings took place, and the first German companies began to use the principle in their products and processes. Particularly noteworthy are the establishment of the “Fuzzy Initiative North Rhine-Westphalia” by the state's Minister of Economic Affairs and the European ELITE Foundation (European Laboratory for Intelligent Techniques Engineering) in Aachen, which pursues the same goals in Europe as the LIFE Institute in Japan. The first products appeared at German trade fairs last year.

The other European countries as well as the USA began to show intense interest in fuzzy technology. Since the middle of 1992 the development in the USA seems to be faster than in Europe. It is to be hoped that the Europeans will succeed in maintaining their lead in the fields of data analysis and expert systems and in reducing the gap in fuzzy control compared to Japan.

From: Spectrum of Science 3/1993, page 90
© Spektrum der Wissenschaft Verlagsgesellschaft mbH

This article is contained in Spectrum of Science 3/1993