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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]]. If we consider fuzzy logic as a formal logic with a semantics and a deduction apparatus, then we can consider it as a new chapter of multi-valued logic. Sometimes in such a case one uses the expression "fuzzy logic in narrow sense" or "formal fuzzy logic".
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| Now, from a semantical point of view fuzzy logic is not too different from first order multi-valued logic. As an example, since in 1996 in the basic book by Chang and Kisler "Continuous model theory" several notions of fuzzy logic where defined in a theoretical setting. Nevertheless the aim of fuzzy logic is to find applications while multi-valued logic originates from philosophical questions, in general.
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| Instead, with respect to the notion of a proof and from the point of view of the deduction apparatus the approach of fuzzy logic is completely new. In fact, in multi-valued logic such an apparatus is devoted to define a deduction operator enabling us to associate every (classical) set of axioms with the related (classical) set of theorems. In many cases one consider sufficient to define a deduction apparatus which is able only to generate the set of valid formulas. From such a point of view the paradigm of the deduction is not different in nature from the one of classical logic. Instead in fuzzy logic the notion of approximate [[reasoning]] is crucial and this notion enables us to associate any fuzzy set of proper axioms with the related fuzzy subset of consequences.
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| Fuzzy logic usually allows for truth values to range (inclusively) between 0 and 1, and in its linguistic form, imprecise concepts like "slightly", "quite" and "very". Specifically, it allows graded membership of an elements in a set.
| | '''Fuzzy control''' is the main success of fuzzy set theory and it is devoted to useful applications. |
| | | The idea is that we can consider IF-THEN rules in which fuzzy quantities are involved. |
| Fuzzy logic is controversial in some circles, despite wide acceptance and a broad track record of successful applications. It is rejected by some [[Control theory|control engineers]] for validation and other reasons, and by some [[statistics|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 set]]s.
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| == Applications ==
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| 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.
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| 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.
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| [[Image:Warm_fuzzy_logic_member_function.gif|center]]
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| In this image, ''cold'', ''warm'', and ''hot'' are functions mapping a temperature scale. A point on that scale has three "[[Logical value|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 hot".
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| == Misconceptions and controversies==
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| ;Fuzzy logic is the same as "imprecise logic".
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| :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 — a concept classical logic cannot easily handle due to the [[principle of bivalence]].
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| ;Fuzzy logic is a new way of expressing probability.
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| [[Degree of truth|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. 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 are based on vague definitions of sets, not randomness.
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| 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 proven a derivation of [[Bayes' theorem]] from the concept of [[fuzzy set|fuzzy subsethood]]. Observe that Zadeh has created a fuzzy alternative to probability, which he calls [[possibility theory]] and which is related with [[Dempster-Shafer theory]].
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| == Examples where fuzzy logic is used ==
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| * Automobile and other vehicle subsystems, such as [[Anti-lock braking system|ABS]] and [[cruise control]] (e.g. Tokyo [[monorail]])
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| * [[Air conditioning|Air conditioners]]
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| * 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
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| * [[Camera]]s
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| * [[Digital image processing]], such as [[edge detection]]
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| * [[Rice cooker]]s
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| * [[Dishwasher]]s
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| * [[Elevator]]s
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| * [[Carbon dioxide enrichment burner's controllers]]
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| * [[Washing machine]]s and other [[home appliance]]s
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| * [[Video game]] [[artificial intelligence]]
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| * [[Language filters]] on [[message boards]] and [[chat rooms]] for filtering out offensive text
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| * [[Pattern recognition]] in [[Remote Sensing]]
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| * Fuzzy logic has also been incorporated into some [[microcontroller]]s and [[microprocessor]]s, for instance, the [[Freescale 68HC12]].
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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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| == Formal fuzzy logic ==
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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 logics ===
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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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| 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. An extension of such a theory to the general case of the L-subsets is proposed in a paper by G. Gerla.
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| 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).
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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]] |
| == Bibliography ==
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| *Ahmad M. Ibrahim'', Introduction to Applied Fuzzy Electronics'', ISBN 0-13-206400-6 | |
| * 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
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| * 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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