Fuzzy-Systems in Computer Science

Fuzzy Set Theory was recognized by some scientists in Germany already at the beginning of the seventies. While control engineers in Great Britain developed the concept of a fuzzy controller and showed that it worked, on the Continent, research was more done in the mathematical areas and in particular in Operations Research where the first institutional working group (European working group of fuzzy sets) was established in 1976. In German universities research was predominantly in mathematical areas. At the Universities of Wuppertal and Mainz, for instance, research was performed in the areas of fuzzy topology and algebra while at the University of Braunschweig in the early eighties research on fuzzy measures and the interface between classical statistics and fuzzy set theory was done. At the Institute of Technology in Aachen (Aix-la-Chapelle) empirical and axiomatic research went on at that time. In the Chair for Operations Research, empirical as well as axiomatic basic research concerning operators and membership functions was started in 1972. This research overlapped with the development of “fuzzy linear programming” and its applications to multi-criteria-analysis and various other areas.

A fuzzy controller can determine its output values either by evaluating a fuzzy look-up table or by interpreting directly its linguistic rule set. Sometimes the second way has advantages. In this case a method has to be derived to store the rule set in the actual runtime environment such that the controller can interprete it sufficiently fast at run-time.

In this article a fundamental solution of this task is presented. The original rule set is transformed by replacing rules with an identical conclusion by one single rule. If necessary, the premise of such an integrated rule is reformulated such that each rule premise has a standard structure — an OR-connection of AND-connections of linguistic values. A standard structure premise is coded as a matrix with integer-valued elements. It is shown in detail how to code a premise and what are the storage requirements of this method.

It is demonstrated how the controller calculates its output values during run-time. For each sample point it determines the truth value of each linguistic rule premise by evaluating its corresponding premise matrix.

Fuzzy control was implemented with the help of standard functional units of an automatization language. This approach integrate the novel technique of fuzzy control into an automatization environment. As an application of the introduced storing method, it is shown how one can realize the presented approach by additional functional units.

For the design of a fuzzy controller it is necessary to choose, besides other parameters, suitable membership functions for the linguistic terms and to determine a rule base. This paper deals with the problem of finding a good rule base — the basis of a fuzzy controller. Consulting experts still is the usual but time-consuming and therefore rather expensive method. Besides, after having designed the controller, one cannot be sure that the rule base will lead to near optimal control. This paper shows how to reduce significantly the period of development (and the costs) of fuzzy controllers with the help of genetic algorithms and, above all, how to engender a rule base which is very close to an optimum solution.

The example of the inverted pendulum is used to demonstrate how a genetic algorithm can be designed for an automatic construction of a rule base.

So this paper does not deal with the tuning of an existing fuzzy controller but with the genetic (re-)production of rules, even without the need for experts. Thus, a program is engendered, consisting of simple “IF… THEN…” instructions.

The results of conventional control theory are not really satisfactory in cases of non-linear systems of high order with uncertainties in parameter and structure. This is mainly due to the necessary reduction in order and simplification of the system. A fuzzy controller, on the other hand is not based on mathematical modells. Consequently, it is especially suitable for technical non-linear processes for which verbal control strategies are known. This paper presents the speed and position control of an elastic two-mass-system with slack. A fuzzy controller is compared to the conventional theory. A systematical fuzzy controller design as well as the robustness of the system concerning uncertainties in parameter and structure are researched.

This paper gives an overview to different concepts of neural fuzzy systems. There are already several approaches to combine neural networks and fuzzy systems, to obtain adaptive systems that can use prior knowledge and that can be interpreted by means of linguistic rules as they are used e.g. in fuzzy controllers. Neural fuzzy models can be divided in two classes: Cooperative models which use neural nets and fuzzy systems separately, and hybrid models which create a new architecture using concepts from both worlds. Several of these approaches are discussed in this paper.

The so-called qualitative and structural fuzzy image analysis are desribed. For image data represented as fuzzy sets a special notion of similarity — the analogy -is used in order to compare the images. The notion of analogy has been previously applied in fuzzy decision making. Subsequently the application examples are given. Finally a comparison is made with neural networks in the field of character recognition.

Application of fuzzy control for obtaining better performance from conventional neural networks is a new area in the field of fuzzy-neural combined systems. Conventional backpropagation algorithm for example can be improved by changing network parameters, based on the empirical knowledge gained by the user. This manual adaptation is effectively replaced by a fuzzy controller that contains the a priori knowledge in form from membership functions and rules. The implemented modified backpropagation algorithm with a fuzzy controller for dynamic adaptation of network parameters is tested with a benchmark data set and two real world problems.

Combinations of neural networks and fuzzy controllers that are known from recent publications are mostly cooperative in nature. This means a neural network is used to learn either fuzzy sets or fuzzy rules, and the results are used to build a conventional fuzzy controller. Hybrid approaches on the other hand try to find a new kind of architecture that unifies neural networks and fuzzy controllers. Some of these approaches have problems when it comes to the interpretation of the learning results. This is especially true, when a pure neural architecture is used.

This paper reviews motivations for introducing fuzzy sets and fuzzy logic to knowledge representation in artificial intelligence. First we consider some areas of successful application of conventional approaches to system analysis. We then discuss limitations of these approaches and the reasons behind these limitations.

We introduce different levels of representation for complex systems and discuss issues of granularity and fuzziness in connection with these representation levels. We make a distinction between decomposable and integrated complex systems and discuss the relevance of this distinction for knowledge representation and reasoning. We also distinguish fuzzy relations between quantities of different granularity within one domain from fuzzy relations between two different domains and discuss the need of considering both in artificial intelligence.

We distinguish methods for describing natural, artificial, and abstract systems and contrast the modeling of system function with the modeling of system behavior in connection with the representation of fuzziness. The paper briefly discusses recent criticism of the fuzzy system approach and concludes with a prospect on soft computing in AI.

FUZZYNEX is an extension of Nexpert Object to make fuzzy inference available. The knowledge is completely represented within Nexpert Object, no access on external files is done during inference. FUZZYNEX allows the construction of hybrid knowledge-based systems by combination of fuzzy and crisp reasoning.

This paper presents the language ACCP which is a probabilistic extension of terminological logics and aims at closing the gap between terminological knowledge representation and uncertainty handling. We present the formal semantics underlying the language ACCP and introduce the probabilistic formalism that is based on classes of probabilities and is realized by means of probabilistic constraints. Besides infering implicitly existent probabilistic relationships, the constraints guarantee terminological and probabilistic consistency. Altogether, the new language ACCP applies to domains where both term descriptions and uncertainty have to be handled.

As with any simple and fruitful mathematical notion also with the notion of fuzzy set a huge amount of theoretical considerations is and can be connected. These theoretical considerations concern basic, foundational aspects of that notions of fuzzy set as well as theoretical problems of specialized fields of applications of fuzzy sets and also the use of fuzzy sets instead of the usual, crisp sets e.g. in mathematical theories etc. This is, besides all applications of fuzzy sets and fuzzy methods, a large field of topics. In any case a field too large to be covered in some more specialized conference (or book). Hence not the whole area of main theoretical research in the fuzzy field can be covered here, and only a few remarks shall be devoted to these topics.

Determining a fuzzy controller is a two step process. First one has to fix the control rules which in general have the form: IF α =A _iTHENβ=B _i ,i= 1,. . . ,Nfor some (perhaps multidimensional) input variable α and output variableβ. And then one has to convert these control rules into a fuzzy relationRand to apply this relation via the compositional rule of inference to any fuzzy input value of or. Ideally the rule inputsA _ishould transform into the rule outputsB _i =A _i oR. Thus, mathematically this second step can be seen as the problem to solve a system of fuzzy relational equation for an unknown fuzzy relation.

Besides these simple, “elementary” types of equations also other, more complicated types of equations have been discussed in fuzzy control theory. The present paper discusses some aspects of how results from the solvability theory of elementary fuzzy relational equations can be extended to discuss also more involved types of relational equations.

Monoidal logic is the a common framework for intuitionistic logic, Lukasiewicz logic and to a ceratin extent for Girard’s commutative logic. Soundness and completeness of the corresponding predicate calculi are verified.

It is shown how fuzzy controllers, in particular the Mamdani and Sugeno controller, can be used to interpolate and approximate control functions, i.e., input-output functions which assign to each input value a real output value.

Although defuzzification is an essential functional part of all fuzzy systems, it is not firmly embedded in fuzzy theory yet.

Proceeding from fuzzy decision theory we define defuzzification as crisp decision under fuzzy constraints and achieve a new theoretical foundation of the defuzzification process. From these theoretical considerations we develop a new class of lucidly customizable defuzzification procedures (constrained decision defuzzification CDD). We develop powerful examples of CDD customization and show that CDD is superior to the standard defuzzification algorithms center of gravity and mean of maxima, represented by the parametric basic defuzzification distribution (BADD), concerning static, dynamic and statistical properties.

There are different mathematical frameworks dealing with uncertainty, vagueness and ambiguity: the probabilistic concept, the concept of a fuzzy set, and the concept of a fuzzy measure. The corresponding measures for the amount of relevant information lead to three types of uncertainty measures: Entropies or measures of information, measures of fuzziness and the uncertainty measures in the mathematical theory of evidence.

One purpose of this paper is to focus on recent results of measures of fuzziness and to give a survey on characterizations of these measures. Moreover, we want to show that certain “total entropies” which consist of a “random part” and a “fuzzy part”, are special cases of a more general information theory, where the entropies are dependent upon the events and the probabilities.

Fuzzy classification, fuzzy diagnosis, and fuzzy data analysis are — besides fuzzy control — the most important application areas of fuzzy logic. In this chapter four practical tasks are presented which can roughly be characterized as technical classification and diagnosis, fuzzy data analysis in chemical model creation, medical object recognition, and decision making support by a life insurance. Neural networks and analytical methods of classical statistics try to find explicitely a classifying function with the help of a sample. The development of knowledge-based systems has stimulated modern constructive approaches like IF-THEN-rules and causal networks. In order to deal with vague observations, vague relationships between features, and/or non-crip classification, these analytical and constructive methods were transfered from crisp numbers to fuzzy sets. The contributions of fuzzy sets to the four applications of this chapter are presented. Some general remarks on the applicability and limitations of fuzzy classification conclude this short introduction to fuzzy classification.

It is always impressive to watch an experienced specialist master difficult and complex situations. The real problems are very different but the characteristics of the action are similar: acquiring a flood of information, concentrating on the essential, evaluating and reaching a purpuseful decision. The expert often cant explain his decision making process. Doubtlessly, his knowledge and experiences about the functional connections are very important. However his ability to compare the present situation with typical examples leads to a strategy in a more effective way. This intellectual process means an immense reduction from a lot of information to a few pattern.

The purpose of this paper is to introduce another application of fuzzy set theory on the field of spectra simulation. Spectra are characterized by a lot of bands. Simulation means to forecast the frequency positions of the bands. To do this, the geometry of the molecule and so-called force constants, reflecting intermoleculare bonds, have to be known. However, the numerical values of the force constants are known only aprroximately and partially in most cases. Since the application of iteration methods like least square methods to determine moleculare force fields is bounded, we will propose an alternative way. Starting from a fuzzy interpretation of uncertainty, we develop a completation of the traditional method to the calculation of vibrational frequencies: the Normal Coordinate Analysis (NCA) The method will be demonstrated and discussed on the example of 3- atomic molecules like water H₂O, hydrogensulfid H₂S and hydrogen-selenium K₂Se.

The term fuzzy elastic matching describes an algorithm based on the combination of heuristic search and fuzzy heuristics. It interrelates two medical objects represented as wireframes or attributed skeletons. Parts of a medical object (i.e. coronary arteries) can be identified considering an abstract model. The fusion of different image modalities (i.e. liver angiography/ tomography) into a normalized one can be achieved using a common element. A new fuzzy algebraic and geometric framework enables us to introduce fuzzyness into object representations to treat their natural unsharpness.

Life insurance takes into account interests of all groups involved; on the one hand the interests of the insurance company and on the other hand the interests of the clients. In this context the contract of insurance can be interpreted as a complex decision process which has to consider the different interests. In this article a method for checking an apply will be proposed. The method is motivated by fuzzifying an outranking approach. In the first step the apply will be prejudged by fuzzy inference and product rules and in a second step the analyzed alternatives of risk minimizing will be put in a hierachical order. The best alternative will be realized in a contract The method is realized in the decision support system FRED (fuzzy preference decision support system) which improves a decision support system (DSS) with a fuzzy knowledge-based element to get a usable tool for checking life insurance applies. The system is developed on a personal computer.