# Fuzzy logic

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### Fuzzy logic

1. 1. Fuzzy logicFuzzy logic is derived from fuzzy set theoty dealing with reasoning that is approximate ratherthan precisely deduced from classical predicate logic. It can be thought as the application sideof fuzzy set theory dealing with well thought out real world expert values for a complexproblem.Degrees of truth are often confused with probabilities. However, they are conceptuallydistinct; fuzzy truth represents membership in vaguely defined sets, not likelihood of someevent or condition. To illustrate the difference, consider this scenario: Bob is in a house withtwo adjacent rooms: the kitchen and the dining room. In many cases, Bobs status within theset of things "in the kitchen" is completely plain: hes either "in the kitchen" or "not in thekitchen". What about when Bob stands in the doorway? He may be considered "partially inthe kitchen". Quantifying this partial state yields a fuzzy set membership. With only his littletoe in the dining room, we might say Bob is 99% "in the kitchen" and 1% "in the diningroom", for instance. No event (like a coin toss) will resolve Bob to being completely "in thekitchen" or "not in the kitchen", as long as hes standing in that doorway. Fuzzy sets are basedon vague definitions of sets, not randomness.Fuzzy logic allows for set membership values between and including 0 and 1, shades of grayas 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 fuzzysets and possibility theory. It was introduced in 1065 by Prof Lotfi Zadeh at the University ofCalifornia, Berkley.Fuzzy logic is controversial in some circles, despite wide acceptance and a broad track recordof successful applications. It is rejected by some control engineers for validation and otherreasons, and by some statisticians who hold that probability is the only rigorous mathematicaldescription of uncertainty. Critics also argue that it cannot be a superset of ordinary set theorysince membership functions are defined in terms of conventional.ApplicationsFuzzy logic can be used to control household appliances such as washing machines (whichsense load size and detergent concentration and adjust their wash cycles accordingly) andrefrigerators.A basic application might characterize subranges of a continuous variable. For instance, atemperature measurement for anti-lock brakes might have several separate membershipfunctions defining particular temperature ranges needed to control the brakes properly. Eachfunction maps the same temperature value to a truth value in the 0 to 1 range. These truthvalues can then be used to determine how the brakes should be controlled.In this image, cold, warm, and hot are functions mapping a temperature scale. A point on thatscale has three "truth values" — one for each of the three functions. For the particulartemperature shown, the three truth values could be interpreted as describing the temperatureas, say, "fairly cold", "slightly warm", and "not hot".A more sophisticated practical example is the use of fuzzy logic in high-performance errorcorrection to improve information reception over a limited-bandwidth communication link
2. 2. affected by data-corrupting noise using turbo codes. The front-end of a decoder produces alikelihood measure for the value intended by the sender (0 or 1) for each bit in the datastream. The likelihood measures might use a scale of 256 values between extremes of"certainly 0" and "certainly 1". Two decoders may analyse the data in parallel, arriving atdifferent likelihood results for the values intended by the sender. Each can then use asadditional data the others likelihood results, and repeats the process to improve the resultsuntil consensus is reached as to the most likely values.Misconceptions and controversiesFuzzy logic is the same as "imprecise logic".Fuzzy logic is not any less precise than any other form of logic: it is an organized andmathematical method of handling inherently imprecise concepts. The concept of "coldness"cannot be expressed in an equation, because although temperature is a quantity, "coldness" isnot. 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 easilyhandle due to the principles of bivalence.Fuzzy logic is a new way of expressing probability.Fuzzy logic and probability refer to different kinds of uncertainty. Fuzzy logic is specificallydesigned to deal with imprecision of facts (fuzzy logic statements), while probability dealswith chances of that happening (but still considering the result to be precise). However, this isa point of controversy. Many staticians are persuaded by the work of Bruno de Finetti thatonly one kind of mathematical uncertainty is needed and thus fuzzy logic is unnecessary. Onthe other hand, Bart Kosko argues that probability is a subtheory of fuzzy logic, as probabilityonly handles one kind of uncertainty. He also claims to have proven a theorem demonstratingthat Bayes´theorem can be derived from the concept of fuzzy subsethood. Lotfi Zadeh, thecreator of fuzzy logic, argues that fuzzy logic is different in character from probability, and isnot a replacement for it. He has created a fuzzy alternative to probability, which he callspossibiliy theory. Other controversial approaches to uncertainty include Dempster-Shafertheory 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 real-world use that reasonabout uncertainty using fuzzy logic. It appears that the limitations of fuzzy logic have notbeen detrimental in control applications because current fuzzy controllers are far simplerthan other knowledge-based systems. In future, the technical limitations of fuzzy logic can beexpected to become important in practice, and work on fuzzy controllers will also encounterseveral problems of scale already known for other knowledge-based systems". Reactions toElkans paper are many and varied, from claims that he is simply mistaken, to others whoaccept that he has identified important limitations of fuzzy logic that need to be addressed bysystem designers. In fact, fuzzy logic wasnt largely used at that time, and today it is used tosolve very complex problems in the AI area. Probably the scalability and complexity of thefuzzy system will depend more on its implementation than on the theory of fuzzy logic.Examples where fuzzy logic is used• Automobile and other vehicle subsystems, such as ABS and cruise control (e.g. Tokyomonorail)• Air conditioners• The Massive engine used in the Lord of the Rings films, which helped show hugescale armies create random, yet orderly movements• Cameras• Digital image processing, such as edge detection
3. 3. • Rice cookers• Dishwashers• Elevartors• Washing machines and other home appliances• Video game artificial intelligence• Massage boards and chat rooms• Fuzzy logic has also been incorporated into some microcontrollers andmicroprocessors, for instance Freescale 68HC12.How fuzzy logic is appliedFuzzy logic usually uses IF/THEN rules, or constructs that are equivalent, such as fuzzyassociattive matrices.Rules are usually expressed in the form:IF variable IS set THEN actionFor example, an extremely simple temperature regulator that uses a fan might look like this:IF temperature IS very cold THEN stop fanIF temperature IS cold THEN turn down fanIF temperature IS normal THEN maintain levelIF temperature IS hot THEN speed up fanNotice 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 asthe minimum, maximum, and complement; when they are defined this way, they are calledthe Zadeh operators, because they were first defined as such in Zadehs original papers. So forthe 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 aset using a mathematical formula.In application, the programming language Prolog is well geared to implementing fuzzy logicwith its facilities to setup a database of "rules" which are queried to deduct logic. This sort ofprogramming is known as logic programming.Other examplesIf a man is 1.8 meters, consider him as tall:IF male IS true AND height >= 1.8 THEN is_tall IS trueIF male IS true AND height >= 1.8 THEN is_short IS falseThe fuzzy rules do not make the sharp distinction between tall and short, that is not sorealistic:IF height <= medium male THEN is_short IS agree somehowIF height >= medium male THEN is_tall IS agree somehowIn the fuzzy case, there are no such heights like 1.83 meters, but there are fuzzy values, likethe following assignments:dwarf male = [0, 1.3] msmall male = (1.3, 1.5]medium male = (1.5, 1.8]tall male = (1.8, 2.0]giant male > 2.0 m
4. 4. For the consequent, there are also not only two values, but five, say:agree not = 0agree little = 1agree somehow = 2agree a lot = 3agree fully = 4In the binary, or "crisp", case, a person of 1.79 meters of height is considered short. If anotherperson is 1.8 meters or 2.25 meters, these persons are considered tall.The crisp example differs deliberately from the fuzzy one. We did not put in the antecedentIF male >= agree somehow AND ...as gender is often considered as a binary information. So, it is not so complex like being tall.Formal fuzzy logicIn mathematical logic, there are several formal systems that model the above notions of"fuzzy logic". Note that they use a different set of operations than above mentioned Zadehoperators.Bibliography Constantin von Altrock, Fuzzy Logic and NeuroFuzzy Applications Explained (2002), Earl Cox, The Fuzzy Systems Handbook (1994), Charles Elkan. The Paradoxical Success of Fuzzy Logic, (1993) Petr Hájek, Metamathematics of fuzzy logic (1998), Frank Höppner, Frank Klawonn, Rudolf Kruse and Thomas Runkler, Fuzzy ClusterAnalysis (1999), George Klir and Tina Folger, Fuzzy Sets, Uncertainty, and Information (1988), George Klir, UTE H. St.Clair and Bo Yuan Fuzzy Set Theory Foundations andApplications (1997), George Klir and Bo Yuan, Fuzzy Sets and Fuzzy Logic (1995) Bart Kosko, Fuzzy Thinking: The New Science of Fuzzy Logic (1993), Ronald Yager and Dimitar Filev, Essentials of Fuzzy Modeling and Control (1994), Hans-Jürgen Zimmermann, Fuzzy Set Theory and its Applications (2001), Kevin M. Passino and Stephen Yurkovich, Fuzzy Control (1998).