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== About Fuzzy Logic ==

{{Wikipedia|Fuzzy logic}}

'''Fuzzy logic''' is derived from [[fuzzy set]] theory dealing with [[reasoning]] that is approximate rather than precisely deduced from classical [[predicate logic (Philosophy)|predicate logic]]. It can be thought of as the application side of fuzzy set theory dealing with well thought out real world expert values for a complex problem (Klir 1997).

[[Degrees of truth]] are often confused with [[probability|probabilities]].  However, they are conceptually distinct; fuzzy truth represents [[Membership function (mathematics)|membership]] in vaguely defined sets, not likelihood of some event or condition. For example, if a 100-ml glass contains 30 ml of water, then, for two fuzzy sets, Empty and Full, one might define the glass as being 0.7 empty and 0.3 full.  Note that the concept of emptiness would be subjective and thus would depend on the observer or designer.  Another designer might equally well design a set membership function where the glass would be considered full for all values down to 50 ml. A probabilistic setting would first define a [[scalar]] variable for the fullness of the glass, and second, conditional distributions describing the probability that someone would call the glass full given a specific fullness level.  Note that the conditioning can be achieved by having a specific observer that randomly selects the label for the glass, a distribution over deterministic observers, or both.  While fuzzy logic avoids talking about randomness in this context, this simplification at the same time obscures what is exactly meant by the statement the 'glass is 0.3 full'.

Fuzzy logic allows for [[Membership function (mathematics)|set membership values]] to range (inclusively) between 0 and 1, 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 [[Lotfi Asker Zadeh|Lotfi Zadeh]] at the [[University of California, Berkeley]].

Fuzzy logic is controversial in some circles and is rejected by some [[Control theory|control engineers]] and by most [[statistics|statisticians]] who hold that [[probability]] is the only rigorous mathematical description of [[uncertainty]].{{Verify source|date=July 2007}} Critics also argue that it cannot be a superset of [[ordinary set theory]] since membership functions are defined in terms of [[conventional set]]s.

== Applications ==
Fuzzy logic can be used to [[control system|control]] [[household appliance]]s such as [[washing machine]]s (which sense load size and [[detergent]] concentration and adjust their wash cycles accordingly) and [[refrigerator]]s.

A basic application might characterize subranges of a continuous variable.  For instance, a temperature measurement for anti-lock 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.

[[Image:Fuzzy logic temperature en.svg|center]]

In this image, ''cold'', ''warm'', and ''hot'' are functions mapping a temperature scale. A point on that scale has three "[[Logical value|truth values]]" &mdash; one for each of the three functions. For the particular temperature illustrated with the vertical line, the three truth values could be interpreted as describing the temperature as, say, "fairly cold" (blue arrow), "slightly warm" (yellow arrow), and "not hot" (red arrow).

== Misconceptions and controversies==

{{POV-section|date=December 2007}}

;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 there is no sharp cutoff between "cold" and "not cold", where something is "cold" at N degrees but "not cold" at N+1 degrees &mdash; a concept classical logic cannot easily handle due to the [[principle of bivalence]]. The result has no set answer so it is believed to be a 'fuzzy' answer.

;Fuzzy logic is a new way of expressing probability. 
:Fuzzy logic and probability are different ways of expressing uncertainty. While both fuzzy logic and probability theory can be used to represent subjective belief, fuzzy set theory uses the concept of fuzzy set membership (i.e. ''how much'' a variable is in a set), probability theory uses the concept of [[subjective probability]] (i.e. ''how probable'' do I think that a variable is in a set). While this distinction is mostly philosophical, the fuzzy-logic-derived [[possibility theory|possibility measure]] is inherently different from the [[probability measure]], hence they are not ''directly'' equivalent. However, 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 proven a derivation of [[Bayes' theorem]] from the concept of [[fuzzy set|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 [[Expert Systems/Dempster-Shafer Theory|Dempster-Shafer theory]] and [[rough set]]s.  

;Fuzzy logic will be difficult to scale to larger problems.
:This criticism is mainly due to the fact that there exist problems with conditional possibility, the fuzzy set theory equivalent of conditional probability (see Halpen (2003), section 3.8). This makes it difficult to perform inference. However there have not been many studies comparing fuzzy-based systems with probabilistic ones.

== Examples where fuzzy logic is used ==

* Automobile and other vehicle subsystems, such as [[automatic transmissions]], [[Anti-lock braking system|ABS]] and [[cruise control]] (e.g. Tokyo [[monorail]])
* [[Air conditioning|Air conditioners]]
* The [[Massive (software)|Massive]] engine used in the ''[[The Lord of the Rings film trilogy|Lord of the Rings]]'' films, which helped show huge scale armies create random, yet orderly movements
* [[Camera]]s
* [[Digital image processing]], such as [[edge detection]]
* [[Rice cooker]]s
* [[Dishwasher]]s
* [[Elevator]]s
* [[Washing machine]]s and other [[home appliance]]s
* [[Video game]] [[artificial intelligence]]
* [[Language filter]]s on [[internet forum|message boards]] and [[chat room]]s for filtering out offensive text
* [[Pattern recognition]] in [[Remote Sensing]]
* Fuzzy logic has also been incorporated into some [[microcontroller]]s and [[microprocessor]]s, for instance, the [[Freescale 68HC12]].

== How fuzzy logic is applied ==
Fuzzy Set Theory defines Fuzzy Operators on Fuzzy Sets. The problem in applying this is that the appropriate Fuzzy Operator may not be known. For this reason, Fuzzy logic usually uses IF/THEN rules, or constructs that are equivalent, such as [[fuzzy associative matrix|fuzzy associative matrices]].

Rules are usually expressed in the form:<br>
IF ''variable'' IS ''set'' THEN ''action''

For example, an extremely simple temperature regulator that uses a fan might look like this:<br>
IF temperature IS very cold THEN stop fan<br>
IF temperature IS cold THEN turn down fan<br>
IF temperature IS normal THEN maintain level<br>
IF temperature IS hot THEN speed up fan<br>

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 [[logical operator|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.

In application, the [[programming language]] [[Prolog]] is well geared to implementing fuzzy logic with its facilities to set up a database of "rules" which are queried to deduct logic.  This sort of programming is known as [[logic programming]].

Once fuzzy relations are defined, it is possible to develop fuzzy [[relational database]]s. The first fuzzy relational database, FRDB, appeared in [[Maria Zemankova|Maria Zemankova's]] dissertation. After, some other models arose like the Buckles-Petry model, the Prade-Testemale Model, the Umano-Fukami model or the GEFRED model by J.M. [[Medina]], M.A. [[Vila]] et al. In the context of fuzzy databases, some fuzzy querying languages have been defined, highlighting the [[SQLf]] by P. [[Bosc]] et al. and the [[FSQL]] by J. [[Galindo]] et al. These languages define some structures in order to include fuzzy aspects in the [[SQL]] statements, like fuzzy conditions, fuzzy comparators, fuzzy constants, fuzzy constraints, fuzzy thresholds, linguistic labels and so on.

=== Other examples ===

* If a man is 1.8 meters, consider him as tall:<br>
IF male IS true AND height >= 1.8 THEN is_tall IS true; is_short IS false

* The fuzzy rules do not make the sharp distinction between tall and short, that is not so realistic:<br>
IF height <= medium male THEN is_short IS agree somewhat<br>
IF height >= medium male THEN is_tall IS agree somewhat<br>

In the fuzzy case, there are no such heights like 1.83 meters, but there are fuzzy values, like the following assignments:

dwarf male = [0, 1.3] m<br>
short male = (1.3, 1.5]<br>
medium male = (1.5, 1.8]<br>
tall male = (1.8, 2.0]<br>
giant male > 2.0 m<br>

For the [[consequent]], there are also not only two values, but five, say:<br>

agree not = 0<br>
agree little = 1<br>
agree somewhat = 2<br>
agree a lot = 3<br>
agree fully = 4<br>

In the binary, or "crisp", case, a person of 1.79 meters of height is considered short. If another person 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 [[antecedent]] 

IF male >= agree somewhat AND ...

as gender is often considered as a binary information. So, it is not so complex as being tall.
<references/>

== Formal fuzzy logic ==
In [[mathematical logic]], there are several [[formal system]]s that model the above notions of "fuzzy logic"; most of them belong among so-called [[t-norm fuzzy logics]]. Note that they use a different set of operations than above mentioned Zadeh operators.

=== Propositional fuzzy logics ===

The most important propositional fuzzy logics are:
* [[Basic propositional fuzzy logic]] BL is an axiomatization of logic where [[conjunction]] is defined by a continuous [[triangular norm|t-norm]], and implication is defined as the residuum of the t-norm. Its [[structure (mathematical logic)|model]]s correspond to [[BL-algebra]]s.
* [[Lukasiewicz fuzzy logic|Łukasiewicz fuzzy logic]] is a special case of basic fuzzy logic where conjunction is Łukasiewicz t-norm. It has the axioms of basic logic plus an axiom of double negation (so it is not [[intuitionistic logic]]), and its models correspond to [[MV-algebra]]s.
* [[Godel fuzzy logic|Gödel fuzzy logic]] is a special case of basic fuzzy logic where conjunction is [[Gödel]] t-norm. It has the axioms of basic logic plus an axiom of idempotence of conjunction, and its models are called [[G-algebra]]s.
* [[Product fuzzy logic]] is a special case of basic fuzzy logic where conjunction is product t-norm. It has the axioms of basic logic plus another axiom, and its models are called [[product algebra]]s.
* [[Monoidal t-norm logic]] MTL is a generalization of basic fuzzy logic BL where conjunction is realized by a ''left''-continuous t-norm. Its models (MTL-algebras) are prelinear commutative bounded integral [[residuated lattice]]s.
* [[Rational Pavelka logic]] is a generalization of [[multi-valued logic]]. It is an extension of Łukasziewicz fuzzy logic with additional constants.
All these logics encompass the traditional [[propositional logic]] (whose models correspond to [[Boolean algebra (structure)|Boolean algebra]]s).

=== Predicate fuzzy logics ===
These extend the above-mentioned fuzzy logics by adding [[universal quantifier|universal]] and [[existential quantifier]]s in a manner similar to the way that [[predicate logic]] is created from [[propositional logic]]. The semantics of the universal resp. existential quantifier in [[t-norm fuzzy logics]] is the [[infimum]] resp. [[supremum]] of the truth degrees of the instances of the quantified subformula.

=== Effectiveness for fuzzy logics ===
The notions of a "decidable subset" and "[[recursively enumerable]] subset" are basic ones for [[classical mathematics]] and [[classical logic]]. Then, the question of a suitable extension of such concepts to fuzzy set theory arises. A first proposal in such a direction was made by E.S. Santos by the notions of ''fuzzy [[Turing machine]]'', ''Markov normal fuzzy algorithm'' and ''fuzzy program''. Successively, L. Biacino and G. Gerla proposed the following definition where ''Ü'' denotes the set of rational numbers in [0,1].
A fuzzy subset μ : ''S'' <math>\rightarrow</math>[0,1] of a set ''S'' is ''recursively enumerable'' if a recursive map ''h'' : ''S''×''N'' <math>\rightarrow</math>''Ü'' exists such that, for every ''x'' in ''S'', the function ''h''(''x'',''n'') is increasing with respect to ''n'' and μ(''x'') = lim ''h''(''x'',''n'').
We say that μ is ''decidable'' if both μ and its complement –μ are recursively enumerable. An extension of such a theory to the general case of the L-subsets is proposed in a paper by G. Gerla.
The proposed definitions are well related with fuzzy logic. Indeed, the following theorem holds true (provided that the deduction apparatus of the fuzzy logic satisfies some obvious effectiveness property). 

'''Theorem.''' Any axiomatizable fuzzy theory is recursively enumerable. In particular, the fuzzy set of logically true formulas is recursively enumerable in spite of the fact that the crisp set of valid formulas is not recursively enumerable, in general. Moreover, any axiomatizable and complete theory is decidable.

It is an open question to give supports for a ''Church thesis'' for fuzzy logic claiming that the proposed notion of recursive enumerability for fuzzy subsets is the adequate one. To this aim, further investigations on the notions of fuzzy grammar and fuzzy Turing machine should be necessary (see for example Wiedermann's paper). Another open question is to start from this notion to find an extension of [[Gödel]]’s theorems to fuzzy logic.

== Bibliography ==
* {{cite book|last=Von Altrock|first=Constantin|title=Fuzzy logic and NeuroFuzzy applications explained|publisher=Prentice Hall PTR|location=Upper Saddle River, NJ|year=1995|isbn=0-13-368465-2}}
* {{cite journal|last=Biacino|first=L.|coauthors=Gerla, G.|year=2002|title=Fuzzy logic, continuity and effectiveness|url=|journal=Archive for Mathematical Logic|issn=0933-5846|volume=41|issue=7|pages=643-667|doi=10.1007/s001530100128}}
* {{cite book|last=Cox|first=Earl|title=The fuzzy systems handbook: a practitioner's guide to building, using, maintaining fuzzy systems|publisher=AP Professional|location=Boston|year=1994|isbn=0-12-194270-8}}
* {{cite journal|last=Elkan|first=C.|coauthors=|year=1994|title=The Paradoxical Success of Fuzzy Logic|url=|journal=IEEE Expert|issn=0885-9000|volume=9|issue=4|pages=3-8|doi=10.1109/64.336150}}. Available from [http://www-cse.ucsd.edu/users/elkan/papers/fuzzy.ps Elkan's home page].
* {{cite journal|last=Gerla|first=Giangiacomo|coauthors=|year=2006|title=Effectiveness and Multivalued Logics|url=|journal=Journal of Symbolic Logic|issn=0022-4812|volume=71|issue=1|pages=137-162|doi=}}
* {{cite book|last=Hájek|first=Petr|title=Metamathematics of fuzzy logic|publisher=Kluwer|location=Dordrecht|year=1998|isbn=0792352386}}
* {{cite journal|last=Hájek|first=Petr|coauthors=|year=1995|title=Fuzzy logic and arithmetical hierarchy|url=|journal=Fuzzy Sets and Systems|issn=0165-0114|volume=3|issue=8|pages=359-363|doi=10.1016/0165-0114(94)00299-M}}
* {{cite book|last=Halpern|first=Joseph Y.|title=Reasoning about uncertainty |publisher=MIT Press|location=Cambridge, Mass|year=2003|isbn=0-262-08320-5}}
* {{cite book|last=Höppner|first=Frank|coauthors=Klawonn, F.; Kruse, R.; Runkler, T.|title=Fuzzy cluster analysis: methods for classification, data analysis and image recognition|publisher=John Wiley|location=New York|year=1999|isbn=0-471-98864-2}}
* {{cite book|last=Ibrahim|first=Ahmad M.|coauthors=|authorlink=|title=Introduction to Applied Fuzzy Electronics|edition=|publisher=Prentice Hall|location=Englewood Cliffs, N.J|year=1997|isbn=0-13-206400-6}}
* {{cite book|last=Klir|first=George J.|coauthors=Folger, Tina A.|authorlink=George Klir|title=Fuzzy sets, uncertainty, and information|publisher=Prentice Hall |location=Englewood Cliffs, N.J|year=1988|isbn=0-13-345984-5}}
* {{cite book|last=Klir|first=George J.|coauthors=St Clair, Ute H.; Yuan, Bo|title=Fuzzy set theory: foundations and applications |publisher=Prentice Hall|location=Englewood Cliffs, NJ|year=1997|isbn=0133410587}}
* {{cite book|last=Klir|first=George J.|coauthors=Yuan, Bo|title=Fuzzy sets and fuzzy logic: theory and applications|publisher=Prentice Hall PTR|location=Upper Saddle River, NJ|year=1995|isbn=0-13-101171-5}}
* {{cite book|last=Kosko|first=Bart|authorlink=Bart Kosko|title=Fuzzy thinking: the new science of fuzzy logic|publisher=Hyperion|location=New York |year=1993|isbn=0-7868-8021-X}}
* {{cite journal|last=Montagna|first=F.|coauthors=|year=2001|title=Three complexity problems in quantified fuzzy logic|url=|journal=Studia Logica|issn=0039-3215|volume=68|issue=1|pages=143-152|doi=10.1023/A:1011958407631}}
* {{cite book|last=Mundici|first=Daniele|coauthors=Cignoli, Roberto; D'Ottaviano, Itala M. L.|title=Algebraic foundations of many-valued reasoning |publisher=Kluwer Academic|location=Dodre|year=1999|isbn=0-7923-6009-5}}
* {{cite book|last=Novák|first=Vilém|coauthorsr=Perfilieva, Irina; Močkoř, Jiří|title=Mathematical principles of fuzzy logic|publisher=Kluwer Academic|location=Dodrecht|year=1999|isbn=0-7923-8595-0}}
* {{cite book|last=Passino|first=Kevin M.|coauthors=Yurkovich, Stephen|title=Fuzzy control|publisher=Addison-Wesley|location=Boston|year=1998|isbn=020118074X}}
* {{cite journal|last=Scarpellini|first=Bruno|coauthors=|year=1962|title=Die Nichaxiomatisierbarkeit des unendlichwertigen Prädikatenkalküls von Łukasiewicz|url=|journal=Journal of Symbolic Logic|issn=0022-4812|volume=27|issue=2|pages=159-170|doi=10.2307/2964111}}
* {{cite journal|last=Wiedermann|first=J.|coauthors=|year=2004|title=Characterizing the super-Turing computing power and efficiency of classical fuzzy Turing machines|url=|journal=Theor. Comput. Sci.|issn=|volume=317|issue=|pages=61-69|doi=}}
* {{cite book|last=Yager|first=Ronald R.|coauthors=Filev, Dimitar P.|title=Essentials of fuzzy modeling and control|publisher=Wiley|location=New York|year=1994|isbn=0-471-01761-2}}
* {{cite journal|last=Zadeh|first=L.A.|coauthors=|year=1968|title=Fuzzy algorithms|url=|journal=Information and Control|issn=0019-9958|volume=12|issue=2|pages=94-102|doi=10.1016/S0019-9958(68)90211-8}}
* {{cite journal|last=Zadeh|first=L.A.|coauthors=|year=1965|title=Fuzzy sets|url=|journal=Information and Control|issn=0019-9958|volume=8|issue=3|pages=338-353|doi=10.1016/S0019-9958(65)90241-X}}
* {{cite paper|author=Zemankova-Leech, M.|title=Fuzzy Relational Data Bases|version=Ph. D. Dissertation|publisher=Florida State University|date=1983}}
* {{cite book|last=Zimmermann|first=H.|title=Fuzzy set theory and its applications|publisher=Kluwer Academic Publishers|location=Boston|year=2001|isbn=0-7923-7435-5}}

== External links ==

'''Additional articles'''
*[http://en.citizendium.org/wiki/Formal_fuzzy_logic Formal fuzzy logic] - detailed article at [[Citizendium]]
*[http://www.scholarpedia.org/article/Fuzzy_Logic Fuzzy Logic] - article at [[Scholarpedia]]
*[http://plato.stanford.edu/entries/logic-fuzzy/ Fuzzy logic] - article at [[Stanford Encyclopedia of Philosophy]] 
*[http://www.fuzzy-logic.com/ch3.htm Fuzzy Logic for "Just Plain Folks"]
*[http://will.dwinnell.com/will/Putting%20Fuzzy%20Logic%20to%20Work.pdf Putting Fuzzy Logic To Work] ''PC AI'' Mar/Apr, 2002 An Introduction to Fuzzy Rules
*[http://blog.peltarion.com/2006/10/25/fuzzy-math-part-1-the-theory Fuzzy Math] - Beginner level introduction to Fuzzy Logic.

'''Links pages'''
*[http://www.lcc.uma.es/~ppgg/FSQL/ Web page about FSQL]: References and links about [[FSQL]]

'''Software & tools'''
*[http://www.fuzzytech.com Free Educational Software and Application Notes]
*[http://www.openfuzzymath.org InrecoLAN FuzzyMath], Fuzzy logic add-in for OpenOffice.org Calc
*[http://jfuzzylogic.sourceforge.net/ JFuzzyLogic: Open Source Fuzzy Logic Package + FCL (sourceforge, java)]
*[http://www.metarule.com Open fuzzy logic based inference engine and data mining web service based on Metarule]
*[http://mbfuzzit.sourceforge.net Open Source Software "mbFuzzIT" (Java)]

'''Tutorials'''
*[http://www.jimbrule.com/fuzzytutorial.html Fuzzy Logic Tutorial]
*[http://www.byondscape.com/ascape.dmb/Gazoot.2004-0315/ Fuzzy logic in your game] - tutorial aimed towards game programming.
*[http://www.answermath.com/fuzzymath.htm Simple test to check how well you understand it]
*[http://videolectures.net/acai05_berthold_fl/ Fuzzy Logic - Video tutorial]

=== Sample applications ===

*[http://jds.fass.org/cgi/reprint/84/2/400.pdf Agriculture]
*[http://www.isprs.org/istanbul2004/comm6/papers/685.pdf GIS]
*[http://web.archive.org/web/20050425024310/http://www.medialab.ntua.gr/medialab/Papers2003/2003-8/8.pdf Image Processing] (in [[Internet Archive]])
*[http://www.cs.wayne.edu/~mdong/papers/paper_fuzzytree.pdf Machine Learning]
*[http://udel.edu/~ebenson/Journal_Articles/Benson_ASAE_2000_Adaptive_Edge_Detection.pdf Machine Vision]
*[http://www3.sympatico.ca/alawnicz/PAGE0212.PDF Medicine]
*[http://agron.scijournals.org/cgi/content/full/94/6/1222 Model Validation]
*[http://eprint.uq.edu.au/archive/00000625/02/paper13.pdf OCR]
*[http://homepages.cae.wisc.edu/~ningyue/fuzzy.pdf Shape Recognition]
*[http://www.ensc.sfu.ca/~ljilja/ENSC833/Projects/chen/presentation.pdf Telecommunications]