Fuzzy logic

Fuzzy logic is an extension of Boolean logic dealing with the concept of partial truth. Whereas classical logic holds that everything can be expressed in binary terms (0 or 1, black or white, yes or no), fuzzy logic replaces boolean truth values with degrees of truth.
Degrees of truth are often confused with probabilities, although they are conceptually distinct, because fuzzy truth represents membership in vaguely defined sets, not likelihood of some event or condition. To illustrate the difference, consider this scenario: Bob is in a house with two adjacent rooms: the kitchen and the dining room. In many cases, Bob's status within the set of things "in the kitchen" is completely plain: he's either "in the kitchen" or "not in the kitchen". What about when Bob stands in the doorway? He may be considered "partially in the kitchen". Quantifying this partial state yields a fuzzy set membership. With only his little toe in the dining room, we might say Bob is 0.99 "in the kitchen", for instance. No event (like a coin toss) will resolve Bob to being completely "in the kitchen" or "not in the kitchen", as long as he's standing in that doorway. Fuzzy sets are based on vague definitions of sets, not randomness.
Fuzzy logic allows for set membership values between and including 0 and 1, shades of gray as well as black and white, and in its linguistic form, imprecise concepts like "slightly", "quite" and "very". Specifically, it allows partial membership in a set. It is related to fuzzy sets and possibility theory. It was introduced in 1965 by Dr. Lotfi Zadeh of Berkeley.
Fuzzy logic is controversial despite wide acceptance: it is rejected by some control engineers for validation and other reasons, and by some statisticians who hold that probability is the only rigorous mathematical description of uncertainty. Critics also argue that it cannot be a superset of ordinary set theory since membership functions are defined in terms of conventional sets.
Contents 
Applications
Fuzzy logic can be used to control household appliances such as washing machines (which sense load size and detergent concentration and adjust their wash cycles accordingly) and refrigerators.
A basic application might characterize subranges of a continuous variable. For instance, a temperature measurement for antilock brakes might have several separate membership functions defining particular temperature ranges needed to control the brakes properly. Each function maps the same temperature value to a truth value in the 0 to 1 range. These truth values can then be used to determine how the brakes should be controlled.
Warm_fuzzy_logic_member_function.gif
In this image, cold, warm, and hot are functions mapping a temperature scale. A point on that scale has three "truth values" — one for each of the three functions. For the particular temperature shown, the three truth values could be interpreted as describing the temperature as, say, "fairly cold", "slightly warm", and "not at all hot".
Misconceptions and controversies
 Fuzzy logic is the same as "imprecise logic".
 Fuzzy logic is not any less precise than any other form of logic: it is an organized and mathematical method of handling inherently imprecise concepts. The concept of "coldness" cannot be expressed in an equation, because although temperature is a quantity, "coldness" is not. However, people have an idea of what "cold" is, and agree that something cannot be "cold" at N degrees but "not cold" at N+1 degrees — a concept classical logic cannot easily handle due to the principle of bivalence.
 Fuzzy logic is a new way of expressing probability.
 Fuzzy logic and probability refer to different kinds of uncertainty. Fuzzy logic is specifically designed to deal with imprecision. However, this is a point of controversy. Many statisticians are persuaded by the work of Bruno de Finetti that only one kind of mathematical uncertainty is needed and thus fuzzy logic is unnecessary. On the other hand, Bart Kosko argues that probability is a subtheory of fuzzy logic, as probability only handles one kind of uncertainty. He also claims to have proved a theorem demonstrating that Bayes' theorem can be derived from the concept of fuzzy subsethood. Lotfi Zadeh, the creator of fuzzy logic, argues that fuzzy logic is different in character from probability, and is not a replacement for it. He has created a fuzzy alternative to probability, which he calls possibility theory. Other controversial approaches to uncertainty include DempsterShafer theory and Rough Sets.
 Fuzzy logic will be difficult to scale to larger problems
 In a widely circulated and highly controversial paper, Charles Elkan in 1993 commented that "...there are few, if any, published reports of expert systems in realworld use that reason about uncertainty using fuzzy logic. It appears that the limitations of fuzzy logic have not been detrimental in control applications because current fuzzy controllers are far simpler than other knowledgebased systems. In future, the technical limitations of fuzzy logic can be expected to become important in practice, and work on fuzzy controllers will also encounter several problems of scale already known for other knowledgebased systems". Reactions to Elkan's paper are many and varied, from claims that he is simply mistaken, to others who accept that he has identified important limitations of fuzzy logic that need to be addressed by system designers.
Examples where fuzzy logic is used
 Automobile subsystems, such as ABS and cruise control
 Air conditioners
 The MASSIVE engine used in the Lord of the Rings films, which helped show huge scale armies create random, yet orderly movements
 Cameras
 Digital image processing, such as edge detection
 Rice cookers
 Dishwashers
 Elevators
 Washing machines and other home appliances.
 Video game artificial intelligence
Fuzzy logic has also been incorporated into some microcontrollers and microprocessors, for instance, the Motorola 68HC12.
How fuzzy logic is applied
Fuzzy logic usually uses IF/THEN rules, or constructs that are equivalent, such as fuzzy associative matrices.
Rules are usually expressed in the form:
IF variable IS set THEN action
For example, an extremely simple temperature regulator that uses a fan might look like this:
IF temperature IS very cold THEN stop fan IF temperature IS cold THEN turn down fan IF temperature IS normal THEN maintain level IF temperature IS hot THEN speed up fan
Notice there is no "ELSE". All of the rules are evaluated, because the temperature might be "cold" and "normal" at the same time to differing degrees.
The AND, OR, and NOT operators of boolean logic exist in fuzzy logic, usually defined as the minimum, maximum, and complement; when they are defined this way, they are called the Zadeh operators, because they were first defined as such in Zadeh's original papers. So for the fuzzy variables x and y:
NOT x = (1  truth(x)) x AND y = minimum(truth(x), truth(y)) x OR y = maximum(truth(x), truth(y))
There are also other operators, more linguistic in nature, called hedges that can be applied. These are generally adverbs such as "very", or "somewhat", which modify the meaning of a set using a mathematical formula.
See also
 Artificial intelligence
 Artificial neural network
 Biologicallyinspired computing
 Combs method
 Control system
 Defuzzification
 Dynamic logic
 Expert system
 Fuzzy associative matrix
 Fuzzy Control Language
 Fuzzy control system
 Fuzzy set
 Machine learning
 Pattern recognition
 Rough set
Bibliography
 Earl Cox, The Fuzzy Systems Handbook (1994), ISBN 0121942708
 Frank Höppner, Frank Klawonn, Rudolf Kruse and Thomas Runkler, Fuzzy Cluster Analysis (1999), ISBN 0471988642
 George Klir and Tina Folger, Fuzzy Sets, Uncertainty, and Information (1988), ISBN 0133459845
 George Klir and Bo Yuan, Fuzzy Sets and Fuzzy Logic (1995) ISBN 0131011715
 Ronald Yager and Dimitar Filev, Essentials of Fuzzy Modeling and Control (1994), ISBN 0471017612
 Charles Elkan. The Paradoxical Success of Fuzzy Logic. November 1993. Available from Elkan's home page (http://www.cse.ucsd.edu/users/elkan/).
External links
 Fuzzy Logic Newsgroup FAQ (http://www2.cs.cmu.edu/Groups/AI/html/faqs/ai/fuzzy/part1/faq.html)
 Simple test to check how well you understand it (http://www.answermath.com/fuzzymath.htm)
 Open Source Software "mbFuzzIT" (Java) (http://mbfuzzit.sourceforge.net)
 Fuzzy Logic Overview (http://www.austinlinks.com/Fuzzy/overview.html)
 Fuzzy Logic Lecture (http://www.ncst.ernet.in/education/apgdst/aifac/aicontent/fuzzy/fuzzy.pdf)
 Fuzzy Logic Introduction (http://www.fpk.tuberlin.de/~anderl/epsilon/fuzzyintro4.pdf)
 Fuzzy Logic for "Just Plain Folks" (http://www.fuzzylogic.com/ch3.htm)
 Stanford Encyclopedia of Philosophy entry (http://plato.stanford.edu/entries/logicfuzzy/)
Sample applications
 Agriculture (http://jds.fass.org/cgi/reprint/84/2/400.pdf)
 Image Processing (http://www.medialab.ntua.gr/medialab/Papers2003/20038/8.pdf)
 Machine Learning (http://oldwww.cs.wayne.edu/%7emdong/papers/paper_fuzzytree.pdf)
 Machine Vision (http://udel.edu/~ebenson/Journal_Articles/Benson_ASAE_2000_Adaptive_Edge_Detection.pdf)
 Medicine (http://www3.sympatico.ca/alawnicz/PAGE0212.PDF)
 OCR (http://eprint.uq.edu.au/archive/00000625/02/paper13.pdf)
 Shape Recognition (http://homepages.cae.wisc.edu/~ningyue/fuzzy.pdf)
 Telecommunications (http://www.ensc.sfu.ca/~ljilja/ENSC833/Projects/chen/presentation.pdf)ar:منطق ضبابي
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