# Fuzzy logic

Fuzzy logic of a temperature control

Fuzzy logic ( English fuzzy , blurred ',' vague ',' undefined ', fuzzy logic , fuzzy theory , fuzzy logic' or 'fuzzy theory') or blur logic is a theory that the pattern recognition for the precise detection of the imprecise ( Zadeh ) was developed, which then modeling of blur of colloquial descriptions of systems should serve, but today mainly in applied fields such as the control technology plays a role.

As a generalization of the two-valued Boolean logic , it allows, for example, the expression of a property - as provided by the so-called hedge expressions "a bit", "fairly", "strong" or "very" in natural language to strengthen or weaken a predicate - numerically as a degree of membership and thus to model the fuzziness of a linguistic expression with mathematical precision.

The fuzzy logic is based on the fuzzy sets (fuzzy sets). The set is not defined, as before, by the objects that are (or are not) elements of this set, but by the degree to which they belong to this set. This is done using membership functions that assign a numerical value to each element as the degree of membership. The new set operations introduced in this way define the operations of an associated logic calculus that allows inference processes to be modeled .

## Historical development

The reflections on a logic of fuzziness go back to ancient Greece. The philosopher Plato already postulated that there was a third area between the terms true and false . This was in complete contrast to his contemporary Aristotle , who based the precision of mathematics on the fact that a statement can only be either true or false .

The concept of the doubled center, coined by Georg Wilhelm Friedrich Hegel , also has references to the modern concept of blurring .

The fuzzy set theory , i.e. fuzzy set theory , was developed in 1965 by Lotfi Zadeh at the University of California, Berkeley . Fuzzy technology took off in Japan in the 1980s with the so-called Japanese fuzzy wave . A historical example is the control of the fully automatic Sendai subway , the first successful large-scale application with fuzzy logic in practice. Later, fuzzy logic was also widely used in entertainment electronics devices. The European fuzzy wave did not come until the mid-1990s, when the fundamental discussions about fuzzy logic subsided.

## Fuzzy set theory

The fuzzy set theory is to be distinguished from the multi-valued logic described by the Polish logician Jan Łukasiewicz in the 1920s . In the narrower sense, so-called fuzzy logic can be interpreted as a multi-valued logic, and in this respect there is a certain proximity to multi-valued logic, for whose truth value of a logical statement numbers from the real unit interval [0, 1] (the real numbers from 0 to 1) can be used. However, Lotfi Zadeh understands the fuzzy set theory as a formalization of indefinite conceptual scopes in the sense of referential semantics , which allows him to indicate the fuzziness of the affiliation of objects as elements of the sets to be defined gradually using numerical values ​​between 0 and 1. This opened up a more extensive, linguistic interpretation of the fuzzy set theory as the basis of a logic of uncertainty. The term fuzzy logic was initially not used by Zadeh, but only later by the linguist George Lakoff , who also teaches at Berkeley , after Joseph Goguen, a doctoral student of Zadeh, introduced a logic of fuzzy terms .

In linguistic semantics today, however, the majority of fuzzy logic is viewed as unsuitable for providing a model for vagueness and similar phenomena in natural language. Instead of assigning a truth value to an indefinite statement, which is a fraction between 0 (false) and 1 (true), the method of supervaluation is preferred, in which the assignment of a classic truth value (0; 1) is postponed because it is still from depends on a parameter that must be substantiated by information from the context. The underlying model is known as a partial logic (which is in clear contrast to multi-valued logics).

## Fuzzy sets

AND-OR-NOT operators for linking membership functions (subsets).

Basis of the fuzzy logic are the so-called fuzzy sets (engl .: fuzzy sets ). In contrast to traditional sets (also called sharp sets in the context of fuzzy logic), in which an element of a given basic set is either contained or not contained, a fuzzy set is not defined by the objects that are elements of this set (or are not), but about the degree to which they belong to this set.

This is done using membership functions μ A : X → [0,1], which assign each element of the definition set X a number from the real-valued interval [0,1] of the target set , which assigns the degree of membership μ A (x) of each element x to the so defined fuzzy set A indicates. This means that every element becomes an element of every fuzzy set, but with different degrees of membership defining a certain subset.

Zadeh explained new set operations which, as operations of a new logic calculus , justify the multi-valued fuzzy logic and identify it as a generalization of the two-valued, classical logic , which it contains as a special case. These operations on fuzzy sets can be defined as on sharp sets, e.g. B. the formation of intersections (AND), union sets (OR) and complement sets (NOT). To model the logical operators of conjunction (AND), disjunction (OR) and negation (NOT) one uses the function classes of the T-norm and T-conorm .

### negation

The negation in fuzzy logic is done by subtracting the input values ​​from 1. So

  NOT(A)=1-A


### Non-exclusive-OR circuit

The adjunction is made by choosing the higher value of the input values. So

  OR(A;B)=A wenn A>B
B wenn A<=B


### AND circuit

The conjunction is made by choosing the lower value of the input values. So

  AND(A;B)=A wenn A<B
B wenn A>=B


### Exclusive-OR circuit

For the disjunction one complements the smaller two values ​​and chooses the smaller of the two. For more than two input values, the result of the last operation is inserted recursively with the next input value. Easier: one takes the difference of the less extreme from the extreme value opposite it. So

  XOR(A;B)=A   wenn A>B und A<(1-B)
1-B wenn A>B und A>=(1-B)
B   wenn B>=A und B<(1-A)
1-A wenn B>=A und B>=(1-A)


## Fuzzy functions

Summaries of individual membership functions result in the fuzzy functions . An example of this is a fuzzy function for a person's age. This could consist of several roof-shaped triangles, which in turn represent different age types and represent membership functions of these individual age types. Each triangle covers an area spanning several years of the age. A person aged 35 would have the following characteristics: young with a rating of 0.75 (that's still a lot), middle age with a rating of 0.25 (that's a bit) and nothing of the other functions. In other words: at 35 you are pretty much young and a little medium . The fuzzy function assigns each age value a membership function that characterizes it.

In many cases, fuzzy functions are generated using tables from statistical surveys. These can also be collected by the application itself, as long as there is feedback, as in the elevator control. It is also practically important to incorporate the experience and intuitions of an expert in the respective field into a fuzzy function, especially when there are no statistical statements at all, for example when it is a completely new system to be described.

However, this triangular shape is by no means mandatory; in general, the values ​​of fuzzy functions can have any shape as long as their function values ​​remain in the interval [0,1]. In practice, however, such triangular functions are often used due to their easy calculability. Trapeze (not necessarily mirror-symmetrical) are still relatively widespread, but semicircles can also be found in some applications. In principle, more than two sections of a fuzzy function can also overlap (but this does not seem to make sense in the example considered here).

### Example of a non-linear fuzzy function

The following sigmoid function is an example of a non-linear membership function :

${\ displaystyle S (x, a, \ delta) = {\ begin {cases} 0 & x \ leq a- \ delta \\ 2 ({\ frac {x-a + \ delta} {2 \ delta}}) ^ {2 } & a- \ delta a + \ delta \ end {cases}}}$

With the shape of the letter S, the curve expresses an increasing affiliation to the quantity described by a value in the value range [0.1]. Depending on the application, a decreasing membership can be expressed by a corresponding Z-curve:

${\ displaystyle Z (x, a, \ delta) = 1-S (x, a, \ delta)}$

The parameter α indicates the turning point of the S curve, the value δ determines the slope of the curve. The larger δ is chosen, the flatter the course of the resulting function becomes.

Fuzzy function for a person's age

The age of a person can be represented as a fuzzy function using this curve:

Age of a person
designation Membership function
very young ${\ displaystyle s_ {0}: \; (1-S (x, 30.30)) ^ {2}}$
young ${\ displaystyle s_ {1}: \; 1-S (x, 30.30)}$
not very young ${\ displaystyle s_ {2}: \; 1- (1-S (x, 30.30)) ^ {2}}$
more or less old ${\ displaystyle s_ {3}: \; {\ sqrt {S (x, 60,30)}}}$
old ${\ displaystyle s_ {4}: \; S (x, 60.30)}$
very old ${\ displaystyle s_ {5}: \; S (x, 60.30) ^ {2}}$

The colloquial modifiers can very , more or less , and not very represented by simple modification of a given function:

• The colloquial reinforcing modifier very can be represented in the form of an increased exponent (in the example ). The result is a steeper curve in comparison to the output function.${\ displaystyle s_ {0} = s_ {1} ^ {2}}$
• The slang modifier more or less can be expressed by using a lower exponent or the square root of a given function ( ). The result is a flatter curve shape compared to the output function.${\ displaystyle s_ {3} = {\ sqrt {s_ {4}}}}$
• The negation of a colloquial expression can be represented by a simple subtraction ( ).${\ displaystyle s_ {2} = 1-s_ {0}}$

According to the use cases, this form of representation involves linguistic variables . Ultimately, a single numerical value is calculated from the individual weighted statements, which is able to express the age in mathematical form. You can then continue working precisely with this value. Many methods are also possible with this so-called defuzzification, the best known (but by far not always the best) is certainly the Center-of-Gravity method, in which the numerical value is weighted according to the mass of the geometric shape of the individual sections of the membership function. Another possibility is to simply take a weighted average of the function values.

## Application examples

Today, fuzzy logic is used in different areas: One of the main applications is fuzzy controllers , e.g. B. in automation technology , medical technology , entertainment electronics , vehicle technology and other areas of control technology in which fuzzy controllers compete with conventional controllers . It is also used in artificial intelligence , in inference systems , in speech recognition and other areas such as electrical safety (quantitative evaluations).

The use of fuzzy logic can be useful when there is no mathematical description of a situation or problem, but only a verbal description. Even if - as almost always - the existing knowledge has gaps or is partially out of date, the use of fuzzy logic is a good way of making a well-founded statement about a current or future system status. Then a mathematical description is obtained from linguistically formulated sentences and rules using fuzzy logic, which can be used in computer systems. It is interesting that the fuzzy logic can also be used to sensibly control (or regulate) systems if a mathematical relationship between the input and output variables of a system cannot be represented - or could only take place with great effort, so that automation is possible expensive or impossible to implement in real time.

Other applications are the regulation of subways , the forecast of future loads in routers , gateways or cellular base stations , the control of automatic transmissions in automobiles, alarm systems for anesthesia , intermediate frequency filters in radios, anti-lock systems for automobiles, fire alarm technology, and the forecast of energy consumption at energy suppliers, AF-coupled multi-field automatic exposure systems and AF prediction in single-lens reflex cameras , to name a few.

Fuzzy logic has also found its way into business applications. An example with a success rate is the Intelligent Claims Assessment (ISP), which insurance companies use to protect themselves against insurance fraud .

## Definition of terms

Not to be confused with fuzzy logic is the fuzzy search , which allows a fuzzy search in databases , for example if the exact spelling of a name or term is not known. Even if the membership values ​​from the interval [0,1] look formally like probability values , uncertainty is fundamentally different from probability. In particular, it should be noted that the sum of the values ​​of two functions that overlap does not have to be 1. It can be equal to 1, but it can also be higher or lower.

## literature

• Benno Biewer: Fuzzy Methods. Practice-relevant calculation models and fuzzy programming languages . Springer, Berlin 1997, ISBN 3-540-61943-7 .
• Christoph Drösser : Fuzzy logic. Methodical introduction to frizzy thinking . Rowohlt, Reinbek near Hamburg 1996, ISBN 3-499-19619-0 .
• Siegfried Gottwald : Fuzzy Sets and Fuzzy Logic. Foundations of Application - from a Mathematical Point of View. Vieweg and Teknea, Braunschweig / Wiesbaden Toulouse 1993.
• Berthold Heinrich [Ed .:] Measuring, controlling, regulating. Elements of automation technology . 8th edition. Vieweg, Wiesbaden 2005, ISBN 3-8348-0006-6 .
• Ulrich Höhle, Stephen Ernest Rodabaugh: Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory. Springer, 1999, ISBN 0-7923-8388-5 .
• Michels, Klawonn, Kruse, Nürnberger: Fuzzy control. Basics, design, analysis . Springer-Verlag, ISBN 3-540-43548-4 .
• George J. Klir, Bo Yuan: Fuzzy Sets and Fuzzy Logic: Theory and Applications. 1995, ISBN 0-13-101171-5 .
• Thomas Kron: Fuzzy logic for sociology . In: Austrian Journal for Sociology , 2005, no. 3, pp. 51–89.
• Thomas Kron, Lars Winter: Fuzzy Systems - Considerations on the vagueness of social systems . In: Soziale Systeme, 2005, no. 2, pp. 370–394.
• Andreas Mayer [u. a.]: Fuzzy Logic. Introduction and guidelines for practical use . Addison-Wesley, Bonn 1993, ISBN 3-89319-443-6 .
• Daniel McNeill et al. Paul Freiberger: Fuzzy Logic. The fuzzy logic conquers the technology . Droemer Knauer, Munich 1994, ISBN 3-426-26583-4 .
• Rodabaugh, SE; Klement, EP (Ed.): Topological and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets. Springer, 2003, ISBN 978-1-4020-1515-1 .
• Carsten Q. Schneider, Claudius Wagemann: Qualitative Comparative Analysis (QCA) and Fuzzy Sets. Barbara Budrich, 2007, ISBN 978-3-86649-068-0 .
• Rudolf Seising: The Fuzzification of Systems. The emergence of the fuzzy set theory and its first applications - its development up to the 1970s . (Boethius: Texts and Treatises on the History of Mathematics and Natural Sciences, Volume 54). Franz Steiner Verlag, Stuttgart 2005, ISBN 3-515-08768-0 .
• Hans-Jürgen Zimmermann: Fuzzy Set Theory and its Applications . 2001, ISBN 0-7923-7435-5 .
• Wolfgang Anthony Eiden: Precise uncertainty - information modeling through fuzzy sets. Ibidem, 2002, ISBN 3-89821-230-0 .
• Magdalena Missler-Behr: Fuzzy-based controlling instruments - development of fuzzy approaches. Wiesbaden 2001, ISBN 3-8244-9049-8 .
• Jürgen Adamy: Fuzzy Logic, Neural Networks and Evolutionary Algorithms. Shaker Verlag, Aachen 2015, ISBN 978-3-8440-3792-0 .

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## Individual evidence

1. Hartmut Heine: Textbook of biological medicine. Basic regulation and extracellular matrix . 4th edition. 2015, ISBN 978-3-8304-7544-6 , pp. 106 .
2. ^ Lutz J. Heinrich, Armin Heinzl, Friedrich Roithmayr: Wirtschaftsinformatik-Lexikon . 7th edition. 2004, ISBN 3-486-27540-2 , pp. 684 .
3. ^ Zadeh, LA : Fuzzy sets. Information and Control, 8 , 1965: 338-353
4. ^ JA Goguen: The logic of inexact concepts . Synthesis 19 (3/4) 1969, pp. 325-373.
5. ^ A classic essay on this topic is: Hans Kamp , Barbara H. Partee : Prototype theory and compositionality. Cognition, 57 (1995), pp. 129-191. For a search for compromise: Uli Sauerland: Vagueness in Language: The Case Against Fuzzy Logic Revisited. In P. Cintula, C. Fermüller, L. Godo, P. Hájek (Eds.): Understanding Vagueness - Logical, Philosophical, and Linguistic Perspectives (Studies in Logic 36). College Publications, London 2011, pp. 185-198.
6. See Kamp & Partee (1995: 148ff.) (See previous footnote). An introductory presentation of this idea, albeit without the term “supervaluation”, can be found in: S. Löbner: Semantik. An introduction. 2nd Edition. de Gruyter, Berlin 2015, Chapter 11.4
7. ^ Siegfried Altmann : Electrical Safety - Quantitative Evaluation Methods. Self-published 2013 and 2015, ISBN 978-3-00-035816-6 , abstracts (German and English) with 105 pages, appendix volume with 56 own publications, advanced volume (applied qualimetry and fuzzy logic) with 115 pages and 26 appendices (contents: http: //profaltmann.24.eu)/  ( Page no longer available , search in web archivesInfo: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice.