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. Just as in fuzzy set theory the set membership values can range (inclusively) between 0 and 1, in fuzzy logic the degree of truth of a statement can range between 0 and 1 and is not constrained to the two truth values {true, false} as in classic predicate logic.[1] And when linguistic variables are used, these degrees may be managed by specific functions, as discussed below.

Fuzzy set theory was first introduced in 1965 by Lotfi Zadeh at the University of California, Berkeley and fuzzy logic was derived from it for use in applications.[2] It has been applied to diverse fields from control theory to artificial intelligence, yet still remains controversial among most statisticians who prefer probabilistic logic and some control engineers who prefer traditional two valued logic.

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[edit] Fuzzy degrees of truth

Both fuzzy degrees of truth and probabilities range between 0 and 1 and hence may seem similar at first. However, they are distinct conceptually; fuzzy truth represents membership in vaguely defined sets, not likelihood of some event or condition as in probability theory. 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'.

[edit] Applying fuzzy 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.

In this image, cold, warm, and hot are 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".

[edit] Fuzzy 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.[3]

A linguistic variable such as age may have a value such as young or its opposite defined as old. However, the great utility of linguistic variables is that they can be modified via linguistic operations on the primary terms. For instance, if young is associated with the value 0.7 then very young is automatically deduced as having the value 0.7 * 0.7 = 0.49. And not very young gets the value (1 - 0.49) , i.e. 0.51.

In this example, the operator very(X) was defined as X * X, however in general these operators may be uniformly, but flexibly defined to fit the application, resulting in a great deal of power for the expression of both rules and fuzzy facts.

[edit] An example of fuzzy reasoning

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 set THEN action

For example, an extremely 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

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 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 databases. The first fuzzy relational database, FRDB, appeared in 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.

[edit] Other examples

  • If a man is 1.8 meters, consider him as tall:

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:

IF height <= medium male THEN is_short IS agree somewhat
IF height >= medium male THEN is_tall IS agree somewhat

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
short male = (1.3, 1.5]
medium male = (1.5, 1.8]
tall male = (1.8, 2.0]
giant male > 2.0 m

For the consequent, there are also not only two values, but five, say:

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

In the binary, or "crisp", case, a person of 1.79 meters of height is considered medium. 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.

[edit] Formal fuzzy logic

In mathematical logic, there are several formal systems 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.

[edit] 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 t-norm, and implication is defined as the residuum of the t-norm. Its models correspond to BL-algebras.
  • 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-algebras.
  • 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-algebras.
  • 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 algebras.
  • 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 lattices.
  • 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 algebras).

[edit] 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.

[edit] 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. 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 \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 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 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.

[edit] Application areas

[edit] Controversies

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 — 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 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 has created a fuzzy alternative to probability, which he calls possibility theory. Other controversial approaches to uncertainty include Dempster-Shafer theory and rough sets.
Fuzzy logic will be 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 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.

[edit] See also

[edit] Notes

  1. ^ Novák, Vilém (1999) Mathematical principles of fuzzy logic Dodrecht: Kluwer Academic. ISBN 0-7923-8595-0
  2. ^ Zadeh, L.A. (1965). "Fuzzy sets", Information and Control 8 (3): 338-­353
  3. ^ Lotfi Zadeh et al. 1996 Fuzzy Sets, Fuzzy Logic, Fuzzy Systems, World Scientific Press, ISBN 9810224214

[edit] 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. 
  • Elkan, C. (1994). "The Paradoxical Success of Fuzzy Logic". IEEE Expert 9 (4): 3-8. doi:10.1109/64.336150. ISSN 0885-9000. . Available from Elkan's home page.
  • Gerla, Giangiacomo (2006). "Effectiveness and Multivalued Logics". Journal of Symbolic Logic 71 (1): 137-162. 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. 
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  • 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. 
  • Montagna, F. (2001). "Three complexity problems in quantified fuzzy logic". Studia Logica 68 (1): 143-152. doi:10.1023/A:1011958407631. ISSN 0039-3215. 
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  • Novák, Vilém (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. 
  • 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. 
  • Wiedermann, J. (2004). "Characterizing the super-Turing computing power and efficiency of classical fuzzy Turing machines". Theor. Comput. Sci. 317: 61-69. 
  • Yager, Ronald R.; Filev, Dimitar P. (1994). Essentials of fuzzy modeling and control. New York: Wiley. ISBN 0-471-01761-2. 
  • 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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