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| By the expression '''Fuzzy logic''' one denotes several topics which are related with the notion of [[fuzzy subset]] defined in [[1965]] by [[Lotfi Asker Zadeh|Lotfi Zadeh]] at the [[University of California, Berkeley]]. Given a nonempty set ''S'', a ''fuzzy subset'' of ''S'' is a map ''s'' from ''S'' into the interval [0,1]. Then elements in [0,1] are interpreted as truth values and, in accordance, for every x in S, the value s(x) is interpreted as a membership degree of x to s. In other words, a fuzzy subset is a characteristic function in which graded truth values are admitted.
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| The notion of fuzzy subset enables us to represent the extension of vague predicates and relations as "big","slow", "near" "similar", ... and so on.
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| Observe that there are two possible interpretations of the word "fuzzy logic". The first one is related with an informal utilization of the notion of fuzzy set and it is devoted to the applications. In such a case should be better the expression "[[fuzzy set theory]]" of "fuzzy logic in board sense". Another interpretation is given in considering fuzzy logic as a chapter of formal logic. In such a case one uses the expression "fuzzy logic in narrow sense" or "formal fuzzy logic".
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| == Fuzzy logic as a chapter of multi-valued logic ==
| | '''Fuzzy control''' is the main success of fuzzy set theory and it is devoted to useful applications. |
| The notion of fuzzy subset enables us to define a semantics for a first order language in which vague predicates are admitted and an n-ary vague predicate is interpreted by an n-ary fuzzy relation. This was done long time by people interested in multi-valued logic, obviously. As an example, since 1996 in the book "Continuous model theory" by Chang and Kisler all the main notions of fuzzy logic where defined in a theoretical setting.
| | The idea is that we can consider IF-THEN rules in which fuzzy quantities are involved. |
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| Now, even if from a semantical point of view fuzzy logic is not different from first order multi-valued logic, in the deduction apparatus appear a basic difference. In fact, in multi-valued logic the deduction operator is a tool to associate every (classical) set of axioms with the related (classical) set of theorems. From such a point of view the paradigm of the deduction in multi-valued logic is not different in nature from the one of classical logic. Instead in fuzzy logic the notion of [[approximate reasoning]] which is based on the one of graded inference rule is crucial. This notion enables us to associate any fuzzy set of proper axioms with the related fuzzy subset of consequences.
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| Onother basic difference is in the origin and in the agenda. Indeed the origin of fuzzy logic is in control and aim of fuzzy logic is to find applications. Instead, multi-valued logic originates from philosophical and theoretical questions, in general.
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| == Fuzzy logic and probability ==
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| Many peoples compare fuzzy logic with probability theory since both refer to the interval [0,1]. However, they are conceptually distinct since we have not confuse a [[degree of truth]] with a [[probability measure]]. 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 big toe in the dining room, we might say Bob is 99% "in the kitchen" and 1% "in the dining room", 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 refers to properties that can be verified in a graded way and not on randomness.
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| == How fuzzy logic is applied ==
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| {{Unreferenced|date=December 2006}}
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| 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]].
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| Rules are usually expressed in the form:<br>
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| IF ''variable'' IS ''set'' THEN ''action''
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| For example, an extremely simple temperature regulator that uses a fan might look like this:<br>
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| IF temperature IS very cold THEN stop fan<br>
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| IF temperature IS cold THEN turn down fan<br>
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| IF temperature IS normal THEN maintain level<br>
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| IF temperature IS hot THEN speed up fan<br>
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| 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.
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| 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:
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| NOT x = (1 - truth(x))
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| x AND y = minimum(truth(x), truth(y))
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| x OR y = maximum(truth(x), truth(y))
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| 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.
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| 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]].
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| Once fuzzy relations are defined, it is possible to develop fuzzy [[relational database]]s. The first fuzzy relational data base, FRDB, appeared in [[Maria Zemankova|Maria Zemankova's]] dissertation.
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| === Other examples ===
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| * If a man is 1.8 meters, consider him as tall:<br>
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| IF male IS true AND height >= 1.8 THEN is_tall IS true; is_short IS false | |
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| * The fuzzy rules do not make the sharp distinction between tall and short, that is not so realistic:<br>
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| IF height <= medium male THEN is_short IS agree somewhat<br>
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| IF height >= medium male THEN is_tall IS agree somewhat<br>
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| In the fuzzy case, there are no such heights like 1.83 meters, but there are fuzzy values, like the following assignments:
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| dwarf male = [0, 1.3] m<br>
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| short male = (1.3, 1.5]<br>
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| medium male = (1.5, 1.8]<br>
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| tall male = (1.8, 2.0]<br>
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| giant male > 2.0 m<br>
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| For the [[consequent]], there are also not only two values, but five, say:<br>
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| agree not = 0<br>
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| agree little = 1<br>
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| agree somewhat = 2<br>
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| agree a lot = 3<br>
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| agree fully = 4<br>
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| 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.
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| The crisp example differs deliberately from the fuzzy one. We did not put in the [[antecedent]]
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| IF male >= agree somewhat AND ...
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| as gender is often considered as a binary information. So, it is not so complex as being tall.
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| <references/>
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| == Different formal fuzzy logics ==
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| In [[mathematical logic]], there are several [[formal system]]s that model the above notions of "fuzzy logic". Note that they use a different set of operations than above mentioned Zadeh operators.
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| === Propositional fuzzy logics ===
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| * [[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.
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| * [[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.
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| * [[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.
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| * [[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.
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| * [[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.
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| * [[Rational Pavelka logic]] is a generalization of [[multi-valued logic]]. It is an extension of Łukasziewicz fuzzy logic with additional constants.
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| All these logics encompass the traditional [[propositional logic]] (whose models correspond to [[Boolean algebra]]s).
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| === Predicate fuzzy logics ===
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| 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]].
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| === Effectiveness for fuzzy set theory ===
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| 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].
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| '''Definition''' 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'').
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| We say that μ is ''decidable'' if both μ and its complement –μ are recursively enumerable.
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| An extension of such a theory to the general case of the L-subsets is proposed in a paper by G. Gerla. In such a paper one refer to the theory of effective domains.
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| === Effectiveness for fuzzy logic ===
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| Define the set Val of valid formulas as the set of formulas assuming constantly value equal to 1. Then it is possible to prove that among the usual first order logics only Goedel logic has a recursively enumerable set of valid formulas. In the case of Lukasiewicz and product logic, for example, Val is not recursively enumerable (see B. Scarpellini, and Belluce, also such a fact was extensively examined in the book of Hajek). Neverthless, from these results we cannot conclude that these logics are not effective and therefore that an axiomatization is not possible. Indeed, if we refer to the just exposed notion of effectiveness for fuzzy sets, then the following theorem holds true (provided that the deduction apparatus of the fuzzy logic satisfies some obvious effectiveness property).
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| '''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.
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| It is an open question to give a support for a Church thesis for fuzzy computability and to give [[Gödel]]’s theorems for fuzzy logic using the notion of recursively enumerable fuzzy subset. To this aim, it is very important to refer to adequate definitions of fuzzy grammar and of fuzzy Turing machine (see for example [[Wiedermann]]'s paper).
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| == See also == | | == See also == |
| * [[Artificial intelligence]]
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| * [[Artificial neural network]]
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| * [[Neuro-fuzzy]]
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| * [[Biologically-inspired computing]]
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| * [[Combs method]]
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| * [[Concept mining]]
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| * [[Control system]]
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| * [[Defuzzification]]
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| * [[Dynamic logic]]
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| * [[Expert system]]
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| * [[Fuzzy subalgebra]] | | * [[Fuzzy subalgebra]] |
| * [[Fuzzy associative matrix]] | | * [[Fuzzy associative matrix]] |
| * [[FuzzyCLIPS]] expert system | | * [[FuzzyCLIPS]] expert system |
| * [[Fuzzy concept]]
| | * [[Fuzzy control]] |
| * [[Fuzzy Control Language]]
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| * [[Fuzzy control system]] | |
| * [[Fuzzy electronics]]
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| * [[Fuzzy set]] | | * [[Fuzzy set]] |
| * [[Machine learning]] | | * [[Multi-valued logic]] |
| | * [[Neuro-fuzzy]] |
| * [[Paradox of the heap]] | | * [[Paradox of the heap]] |
| * [[Pattern recognition]] | | * [[Pattern recognition]] |
| * [[Rough set]] | | * [[Rough set]] |
| | | * [[Soft-computing]][[Category:Suggestion Bot Tag]] |
| == Bibliography ==
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| *Ahmad M. Ibrahim'', Introduction to Applied Fuzzy Electronics'', ISBN 0-13-206400-6
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| * Von Altrock C., ''Fuzzy Logic and NeuroFuzzy Applications Explained'' (2002), ISBN 0-13-368465-2
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| * Biacino L., Gerla G., Fuzzy logic, continuity and effectiveness, ''Archive for Mathematical Logic'', 41, (2002), 643-667.
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| * Chang C. C.,Keisler H. J., ''Continuous Model Theory'', Princeton University Press, Princeton, 1996.
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| * Cignoli R., D’Ottaviano I. M. L. , Mundici D. , ‘’Algebraic Foundations of Many-Valued Reasoning’’. Kluwer, Dordrecht, 1999.
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| * Cox E., ''The Fuzzy Systems Handbook'' (1994), ISBN 0-12-194270-8
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| * Elkan C.. ''The Paradoxical Success of Fuzzy Logic''. November 1993. Available from [http://www.cse.ucsd.edu/users/elkan/ Elkan's home page].
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| * Hájek P., ''Metamathematics of fuzzy logic''. Kluwer 1998.
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| * Hájek P., Fuzzy logic and arithmetical hierarchy, ''Fuzzy Sets and Systems'', 3, (1995), 359-363.
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| * Höppner F., Klawonn F., Kruse R. and Runkler T., ''Fuzzy Cluster Analysis'' (1999), ISBN 0-471-98864-2.
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| * Klir G. and Folger T., ''Fuzzy Sets, Uncertainty, and Information'' (1988), ISBN 0-13-345984-5.
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| * Klir G. , UTE H. St. Clair and Bo Yuan ''Fuzzy Set Theory Foundations and Applications'',1997.
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| * Klir G. and Bo Yuan, ''Fuzzy Sets and Fuzzy Logic'' (1995) ISBN 0-13-101171-5
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| * [[Bart Kosko]], ''Fuzzy Thinking: The New Science of Fuzzy Logic'' (1993), Hyperion. ISBN 0-7868-8021-X | |
| * Gerla G., Effectiveness and Multivalued Logics, ''Journal of Symbolic Logic'', 71 (2006) 137-162.
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| * Montagna F., Three complexity problems in quantified fuzzy logic. ''Studia Logica'', 68,(2001), 143-152.
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| * Novák V., Perfilieva I, Mockor J., Mathematical Principles of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, (1999).
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| * Scarpellini B., Die Nichaxiomatisierbarkeit des unendlichwertigen Prädikatenkalküls von Łukasiewicz, ''J. of Symbolic Logic'', 27,(1962), 159-170.
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| * Yager R. and Filev D., ''Essentials of Fuzzy Modeling and Control'' (1994), ISBN 0-471-01761-2
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| * Zimmermann H., ''Fuzzy Set Theory and its Applications'' (2001), ISBN 0-7923-7435-5.
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| * Kevin M. Passino and Stephen Yurkovich, ''Fuzzy Control'', Addison Wesley Longman, Menlo Park, CA, 1998.
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| * Wiedermann J. , Characterizing the super-Turing computing power and efficiency of classical fuzzy Turing machines, ''Theor. Comput. Sci.'' 317, (2004), 61-69.
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| * Zadeh L.A., Fuzzy algorithms, ''Information and Control'', 5,(1968), 94-102.
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| * Zadeh L.A., Fuzzy Sets, ‘’Information and Control’’, 8 (1965) 338353.
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| * Zemankova-Leech, M., ''Fuzzy Relational Data Bases'' (1983), Ph. D. Dissertation, Florida State University.
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| [[category:CZ Live]]
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| [[category:Computers Workgroup]]
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| [[category:Mathematics Workgroup]]
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