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fuzzy logic

 
Dictionary: fuzzy logic

n.
A form of algebra employing a range of values from "true" to "false" that is used in decision-making with imprecise data, as in artificial intelligence systems.


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Logic based on the concept of fuzzy sets, in which membership is expressed in varying probabilities or degrees of truth — that is, as a continuum of values ranging from 0 (does not occur) to 1 (definitely occurs). As additional data are gathered, many fuzzy-logic systems are able to adjust the probability values assigned to different parameters. Because some such systems appear able to learn from their mistakes, they are often considered a crude form of artificial intelligence. The term and concept date from a 1965 paper by Lotfi A. Zadeh (born 1921). Fuzzy-logic systems achieved commercial application in the early 1990s. Advanced clothes-washing machines, for example, use fuzzy-logic systems to detect and adapt to patterns of water movement during a wash cycle, increasing efficiency and reducing water consumption. Other products using fuzzy logic include camcorders, microwave ovens, and dishwashers. Other applications include expert systems, self-regulating industrial controls, and computerized speech- and handwriting-recognition programs.

For more information on fuzzy logic, visit Britannica.com.

A mathematical technique for dealing with imprecise data and problems that have many solutions rather than one. Although it is implemented in digital computers which ultimately make only yes-no decisions, fuzzy logic works with ranges of values, solving problems in a way that more resembles human logic.

Fuzzy logic is used for solving problems with expert systems and real-time systems that must react to an imperfect environment of highly variable, volatile or unpredictable conditions. It "smoothes the edges" so to speak, circumventing abrupt changes in operation that could result from relying on traditional either-or and all-or-nothing logic.

A Matter of Degree

The concept was conceived in 1964 by Lotfi Zadeh, former chairman of the electrical engineering and computer science department at the University of California at Berkeley, while he was contemplating how to program software for handwriting recognition. Zadeh expanded on traditional set theory by making membership in a set a matter of degree rather than a yes-no situation. See set theory.

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Business Dictionary: Fuzzy Logic
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In computer Artificial Intelligence, a system of computer instructions enabling the computer to deal with ambiguities. The instructions are not restricted to ‘either/or' choices. Fuzzy logic emulates the way humans think, so its decisions appear to be more natural.

 
Columbia Encyclopedia: fuzzy logic
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fuzzy logic, a multivalued (as opposed to binary) logic developed to deal with imprecise or vague data. Classical logic holds that everything can be expressed in binary terms: 0 or 1, black or white, yes or no; in terms of Boolean algebra, everything is in one set or another but not in both. Fuzzy logic allows for partial membership in a set, values between 0 and 1, shades of gray, and maybe-it introduces the concept of the "fuzzy set." When the approximate reasoning of fuzzy logic is used with an expert system, logical inferences can be drawn from imprecise relationships. Fuzzy logic theory was developed by Lofti A. Zadeh at the Univ. of California in the mid 1960s. However, it was not applied commercially until 1987 when the Matsushita Industrial Electric Co. used it in a shower head that controlled water temperature. Fuzzy logic is now used to optimize automatically the wash cycle of a washing machine by sensing the load size, fabric mix, and quantity of detergent and has applications in the control of passenger elevators, household appliances, cameras, automobile subsystems, and smart weapons.

Bibliography

See L. A. Zadeh, Fuzzy Logic for the Management of Uncertainty (1992); D. McNeill and P. Freiberger, Fuzzy Logic (1993); B. Kosko, Fuzzy Thinking: The New Science of Fuzzy Logic (1993); R. R. Yager and D. P. Filey, Essentials of Fuzzy Modeling and Control (1995).


Wikipedia: Fuzzy logic
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Fuzzy logic is a form of multi-valued logic derived from fuzzy set theory to deal with reasoning that is approximate rather than precise. In contrast with "crisp logic", where binary sets have binary logic, the fuzzy logic variables may have a membership value of not only 0 or 1 – that is, the degree of truth of a statement can range between 0 and 1 and is not constrained to the two truth values of classic propositional logic.[1] Furthermore, when linguistic variables are used, these degrees may be managed by specific functions.

Fuzzy logic emerged as a consequence of the 1965 proposal of fuzzy set theory by Lotfi Zadeh.[2][3] Though fuzzy logic has been applied to many fields, from control theory to artificial intelligence, it still remains controversial among most statisticians, who prefer Bayesian logic, and some control engineers, who prefer traditional two-valued logic.

Contents

Degrees of truth

Both degrees of truth and probabilities range between 0 and 1 and hence may seem similar at first. However, they are distinct conceptually; truth represents membership in vaguely defined sets, not likelihood of some event or condition as in probability theory. For example, let a 100 ml glass contain 30 ml of water. Then we may consider two concepts: Empty and Full. The meaning of each of them can be represented by a certain fuzzy set. Then 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. It is essential to realize that fuzzy logic uses truth degrees as a mathematical model of the vagueness phenomenon while probability is a mathematical model of randomness. 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. This model, however, has no sense without accepting occurrence of some event, e.g. that after a few minutes, the glass will be half empty. 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. Consequently, probability has nothing in common with fuzziness, these are simply different concepts which superficially seem similar because of using the same interval of real numbers [0, 1]. Still, since theorems such as De Morgan's have dual applicability and properties of random variables are analogous to properties of binary logic states, one can see where the confusion might arise.

Applying truth values

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.

Fuzzy logic temperature

In this image, the meaning of the expressions cold, warm, and hot is represented by functions mapping a temperature scale. A point on that scale has three "truth values" — one for each of the three functions. The vertical line in the image represents a particular temperature that the three arrows (truth values) gauge. Since the red arrow points to zero, this temperature may be interpreted as "not hot". The orange arrow (pointing at 0.2) may describe it as "slightly warm" and the blue arrow (pointing at 0.8) "fairly cold".

Linguistic variables

While variables in mathematics usually take numerical values, in fuzzy logic applications, the non-numeric linguistic variables are often used to facilitate the expression of rules and facts.[4]

A linguistic variable such as age may have a value such as young or its antonym old. However, the great utility of linguistic variables is that they can be modified via linguistic hedges applied to primary terms. The linguistic hedges can be associated with certain functions. For example, L. A. Zadeh proposed to take the square of the membership function. This model, however, does not work properly. For more details, see the references.

Example

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 matrices.

Rules are usually expressed in the form:
IF variable IS property THEN action

For example, a 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

There is no "ELSE" – all of the rules are evaluated, because the temperature might be "cold" and "normal" at the same time to different 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. 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[citation needed] 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 databases. The first fuzzy relational database, FRDB, appeared in Maria Zemankova's dissertation. Later, 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.

Mathematical fuzzy logic

In mathematical logic, there are several formal systems of "fuzzy logic"; most of them belong among so-called t-norm fuzzy logics.

Propositional fuzzy logics

The most important propositional fuzzy logics are:

  • Monoidal t-norm-based propositional fuzzy logic MTL is an axiomatization of logic where conjunction is defined by a left continuous t-norm, and implication is defined as the residuum of the t-norm. Its models correspond to MTL-algebras that are prelinear commutative bounded integral residuated lattices.
  • Basic propositional fuzzy logic BL is an extension of MTL logic where conjunction is defined by a continuous t-norm, and implication is also defined as the residuum of the t-norm. Its models correspond to BL-algebras.
  • Łukasiewicz fuzzy logic is the extension of basic fuzzy logic BL where standard conjunction is the Łukasiewicz t-norm. It has the axioms of basic fuzzy logic plus an axiom of double negation, and its models correspond to MV-algebras.
  • Gödel fuzzy logic is the extension of basic fuzzy logic BL where conjunction is Gödel t-norm. It has the axioms of BL plus an axiom of idempotence of conjunction, and its models are called G-algebras.
  • Product fuzzy logic is the extension of basic fuzzy logic BL where conjunction is product t-norm. It has the axioms of BL plus another axiom for cancellativity of conjunction, and its models are called product algebras.
  • Fuzzy logic with evaluated syntax (sometimes also called Pavelka's logic), denoted by EVŁ, is a further generalization of mathematical fuzzy logic. While the above kinds of fuzzy logic have traditional syntax and many-valued semantics, in EVŁ is evaluated also syntax. This means that each formula has an evaluation. Axiomatization of EVŁ stems from Łukasziewicz fuzzy logic. A generalization of classical Gödel completeness theorem is provable in EVŁ.

Predicate fuzzy logics

These extend the above-mentioned fuzzy logics by adding universal and existential quantifiers 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.

Higher-order fuzzy logics

These logics, called fuzzy type theories, extend predicate fuzzy logics to be able to quantify also predicates and higher order objects. A fuzzy type theory is a generalization of classical simple type theory introduced by B. Russell [5] and mathematically elaborated by A. Church [6] and L. Henkin[7].

Decidability issues for fuzzy logic

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 (see Santos 1970). Successively, L. Biacino and G. Gerla showed that such a definition is not adequate and therefore proposed the following one. Ü denotes the set of rational numbers in [0,1]. A fuzzy subset s : S \rightarrow[0,1] of a set S is recursively enumerable if a recursive map h : S×N \rightarrowÜ exists such that, for every x in S, the function h(x,n) is increasing with respect to n and s(x) = lim h(x,n). We say that s is decidable if both s and its complement –s are recursively enumerable. An extension of such a theory to the general case of the L-subsets is proposed in Gerla 2006. 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.

Application areas

Fuzzy logic is used in the operation or programming of:

Objections to fuzzy logic

Identical to "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 — 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 simply provides a mathematical model of the vagueness which is manifested in the above example.

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 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 subsethood. Lotfi Zadeh argues that fuzzy logic is different in character from probability, and is not a replacement for it. He fuzzified probability to fuzzy probability and also generalized it to what is called possibility theory. Other approaches to uncertainty include Dempster-Shafer theory and rough sets.
Note, however, that fuzzy logic is not controversial to probability but rather complementary (cf. [8])

Difficult to scale to larger problems

This criticism is mainly because there exist problems with conditional possibility, the fuzzy set theory equivalent of conditional probability (see Halpern (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.

See also

Notes

  1. ^ Novák, V., Perfilieva, I. and Močkoř, J. (1999) Mathematical principles of fuzzy logic Dodrecht: Kluwer Academic. ISBN 0-7923-8595-0
  2. ^ "Fuzzy Logic". Stanford Encyclopedia of Philosophy. Stanford University. 2006-07-23. http://plato.stanford.edu/entries/logic-fuzzy/. Retrieved 2008-09-29. 
  3. ^ Zadeh, L.A. (1965). "Fuzzy sets", Information and Control 8 (3): 338–353.
  4. ^ Zadeh, L. A. et al. 1996 Fuzzy Sets, Fuzzy Logic, Fuzzy Systems, World Scientific Press, ISBN 9810224214
  5. ^ Russell, B. Mathematical logic as based on the theory of types, American Journal of Mathematics 30 (1908) 222-262.
  6. ^ Church, A. A formulation of the simple theory of types, J. Symb. Logic 5 (1940) 56--68.
  7. ^ Henkin, L. Completeness in the theory of types, J. Symb. Logic 15 (1950) 81-91.
  8. ^ Novák, V. Are fuzzy sets a reasonable tool for modeling vague phenomena?, Fuzzy Sets and Systems 156 (2005) 341--348.

Bibliography

  • Von Altrock, Constantin (1995). Fuzzy logic and NeuroFuzzy applications explained. Upper Saddle River, NJ: Prentice Hall PTR. ISBN 0-13-368465-2. 
  • Biacino, L.; Gerla, G. (2002). "Fuzzy logic, continuity and effectiveness". Archive for Mathematical Logic 41 (7): 643–667. doi:10.1007/s001530100128. ISSN 0933-5846. 
  • Cox, Earl (1994). The fuzzy systems handbook: a practitioner's guide to building, using, maintaining fuzzy systems. Boston: AP Professional. ISBN 0-12-194270-8. 
  • Gerla, Giangiacomo (2006). "Effectiveness and Multivalued Logics". Journal of Symbolic Logic 71 (1): 137–162. doi:10.2178/jsl/1140641166. ISSN 0022-4812. 
  • Hájek, Petr (1998). Metamathematics of fuzzy logic. Dordrecht: Kluwer. ISBN 0792352386. 
  • Hájek, Petr (1995). "Fuzzy logic and arithmetical hierarchy". Fuzzy Sets and Systems 3 (8): 359–363. doi:10.1016/0165-0114(94)00299-M. ISSN 0165-0114. 
  • Halpern, Joseph Y. (2003). Reasoning about uncertainty. Cambridge, Mass: MIT Press. ISBN 0-262-08320-5. 
  • Höppner, Frank; Klawonn, F.; Kruse, R.; Runkler, T. (1999). Fuzzy cluster analysis: methods for classification, data analysis and image recognition. New York: John Wiley. ISBN 0-471-98864-2. 
  • Ibrahim, Ahmad M. (1997). Introduction to Applied Fuzzy Electronics. Englewood Cliffs, N.J: Prentice Hall. ISBN 0-13-206400-6. 
  • Klir, George J.; Folger, Tina A. (1988). Fuzzy sets, uncertainty, and information. Englewood Cliffs, N.J: Prentice Hall. ISBN 0-13-345984-5. 
  • Klir, George J.; St Clair, Ute H.; Yuan, Bo (1997). Fuzzy set theory: foundations and applications. Englewood Cliffs, NJ: Prentice Hall. ISBN 0133410587. 
  • Klir, George J.; Yuan, Bo (1995). Fuzzy sets and fuzzy logic: theory and applications. Upper Saddle River, NJ: Prentice Hall PTR. ISBN 0-13-101171-5. 
  • Kosko, Bart (1993). Fuzzy thinking: the new science of fuzzy logic. New York: Hyperion. ISBN 0-7868-8021-X. 
  • Kosko, Bart (July 1993). "Fuzzy Logic". Scientific American 269 (1): 76–81. 
  • Montagna, F. (2001). "Three complexity problems in quantified fuzzy logic". Studia Logica 68 (1): 143–152. doi:10.1023/A:1011958407631. ISSN 0039-3215. 
  • Mundici, Daniele; Cignoli, Roberto; D'Ottaviano, Itala M. L. (1999). Algebraic foundations of many-valued reasoning. Dodrecht: Kluwer Academic. ISBN 0-7923-6009-5. 
  • Novák, Vilém (1989). Fuzzy Sets and Their Applications. Bristol: Adam Hilger. ISBN 0-85274-583-4. 
  • Novák, Vilém (2005). "On fuzzy type theory". Fuzzy Sets and Systems 149: 235–273. doi:10.1016/j.fss.2004.03.027. 
  • Novák, Vilém; Perfilieva, Irina; Močkoř, Jiří (1999). Mathematical principles of fuzzy logic. Dodrecht: Kluwer Academic. ISBN 0-7923-8595-0. 
  • Passino, Kevin M.; Yurkovich, Stephen (1998). Fuzzy control. Boston: Addison-Wesley. ISBN 020118074X. 
  • Pu, Pao Ming; Liu, Ying Ming (1980), "Fuzzy topology. I. Neighborhood structure of a fuzzy point and Moore-Smith convergence", Journal of Mathematical Analysis and Applications 76 (2): 571–599, doi:10.1016/0022-247X(80)90048-7, ISSN 0022-247X 
  • Santos, Eugene S. (1970). "Fuzzy Algorithms". Information and Control 17 (4): 326-339. 
  • Scarpellini, Bruno (1962). "Die Nichaxiomatisierbarkeit des unendlichwertigen Prädikatenkalküls von Łukasiewicz". Journal of Symbolic Logic 27 (2): 159–170. doi:10.2307/2964111. ISSN 0022-4812. 
  • Steeb, Willi-Hans (2008). The Nonlinear Workbook: Chaos, Fractals, Cellular Automata, Neural Networks, Genetic Algorithms, Gene Expression Programming, Support Vector Machine, Wavelets, Hidden Markov Models, Fuzzy Logic with C++, Java and SymbolicC++ Programs: 4edition. World Scientific. ISBN 981-281-852-9. 
  • Wiedermann, J. (2004). "Characterizing the super-Turing computing power and efficiency of classical fuzzy Turing machines". Theor. Comput. Sci. 317: 61–69. doi:10.1016/j.tcs.2003.12.004. 
  • Yager, Ronald R.; Filev, Dimitar P. (1994). Essentials of fuzzy modeling and control. New York: Wiley. ISBN 0-471-01761-2. 
  • Van Pelt, Miles (2008). Fuzzy Logic Applied to Daily Life. Seattle, WA: No No No No Press. ISBN 0-252-16341-9. 
  • Wilkinson, R.H. (1963). "A method of generating functions of several variables using analog diode logic". IEEE Transactions on Electronic Computers 12: 112–129. doi:10.1109/PGEC.1963.263419. 
  • Zadeh, L.A. (1968). "Fuzzy algorithms". Information and Control 12 (2): 94–102. doi:10.1016/S0019-9958(68)90211-8. ISSN 0019-9958. 
  • Zadeh, L.A. (1965). "Fuzzy sets". Information and Control 8 (3): 338-­353. doi:10.1016/S0019-9958(65)90241-X. ISSN 0019-9958. 
  • Zemankova-Leech, M. (1983). Fuzzy Relational Data Bases. Ph. D. Dissertation. Florida State University. 
  • Zimmermann, H. (2001). Fuzzy set theory and its applications. Boston: Kluwer Academic Publishers. ISBN 0-7923-7435-5. 

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