Difference between revisions of "Fuzzy logic" - New World Encyclopedia

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'''Fuzzy logic''' is derived from [[fuzzy set|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).
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'''Fuzzy logic''', when construed in a wider sense, is the theory of ''fuzzy sets''. Given a set ''U'', fuzzy sets are mappings from ''U'' into the unit interval [0, 1]. The value that a given fuzzy set ''A'' assigns to an element ''x'' in ''U'' is called the degree of the membership of ''x'' in the fuzzy set ''A'', and, in this sense, fuzzy sets are regarded as generalizing the concept of sets in [[ordinary set theory]].  
  
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Degrees of truth are often confused with [[probability|probabilities]].  However, they are conceptually distinct; fuzzy truth represents 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 little 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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The concept of fuzzy sets provides a convenient way to represent various notions with imprecision, vagueness, or fuzziness, e.g. young, tall, cold, etc., which we frequently employ in our everyday life. As such, fuzzy logic has been applied for various engineering purposes.
  
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Fuzzy logic allows for [[Membership function (mathematics)|set membership values]] between and including 0 and 1, shades of gray as well as black and white, 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 Prof. [[Lotfi Zadeh]] at the [[University of California, Berkeley]].
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When construed in a narrower sense, fuzzy logic is an extension of ordinary two-valued logics in such a way that the points in the unit interval are allowed as truth-values. As the truth-values are generalized in such a way, usual truth-functional operations are generalized accordingly.  
  
 
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.
 
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 Sets ==
 +
Fuzzy logic studies fuzzy sets, which was first introduced by L. Zadeh in 1965. Zadeh maintains that the meanings of many words in natural language come with degrees. 12 years old and 18 years old are clearly both young; however 12 years old is younger than 18 years old. To represent this, he introduces the concept of ''fuzzy subsets''. A fuzzy subset of a given set ''U'' is a function from ''U'' into [1, 0]. The value that a given fuzzy set ''A'' assigns to an element ''x'' in ''U'' is called ''the degree of the membership of x in the fuzzy set A''. Fuzzy subsets are usually referred to simply as ''fuzzy sets''. Using this framework, the meaning of, say, the word “young” can be represented. Take the set of natural numbers and define some fuzzy set, as you like, that assigns values in the unit interval to natural numbers so that, say, 12 (years old) get some value (e.g. .95) higher than the value that 18 (e.g. .85). In that case, the value that each number gets assigned represents the degree of youth. The degree of the membership of 12 in the “youth” subset is higher than that of 18. 
 +
 
 +
This concept of fuzzy sets generalizes the concept of sets in ordinary set theory. Given a set ''U'', a subset ''S'', in the ordinary sense, of ''U'' are determined by a function from ''U'' to {1, 0}. The elements of ''U'' that get 1 assigned represent the elements in ''S'' and the elements that get 0 assigned represent the elements that are not in ''S''. The elements of ''U'' are all either in, or not in, the subset. However, fuzzy subsets are allowed to take any value in the unit interval other than just 1 and 0. In this sense, the sets in the ordinary sense are special cases of fuzzy sets.
 +
 
 +
==Applications in Engineering==
 +
 
 +
===Examples===
 
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.
 
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 more sophisticated practical example is the use of fuzzy logic in high-performance [[error-correcting code|error correction]] to improve information reception over a limited-bandwidth communication link affected by data-corrupting noise using [[turbo code]]s. The front-end of a decoder produces a likelihood measure for the value intended by the sender (0 or 1) for each bit in the data stream. The likelihood measures might use a scale of 256 values between extremes of "certainly 0" and "certainly 1". Two decoders may analyse the data in parallel, arriving at different likelihood results for the values intended by the sender. Each can then use as additional data the other's likelihood results, and repeats the process to improve the results until consensus is reached as to the most likely values.
 
A more sophisticated practical example is the use of fuzzy logic in high-performance [[error-correcting code|error correction]] to improve information reception over a limited-bandwidth communication link affected by data-corrupting noise using [[turbo code]]s. The front-end of a decoder produces a likelihood measure for the value intended by the sender (0 or 1) for each bit in the data stream. The likelihood measures might use a scale of 256 values between extremes of "certainly 0" and "certainly 1". Two decoders may analyse the data in parallel, arriving at different likelihood results for the values intended by the sender. Each can then use as additional data the other's likelihood results, and repeats the process to improve the results until consensus is reached as to the most likely values.
  
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== Misconceptions and controversies==
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=== Other Examples ===
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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 something cannot be "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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:Fuzzy logic and probability refer to different kinds of uncertainty. Fuzzy logic is specifically designed to deal with '''imprecision''' of facts (fuzzy logic statements), while probability deals with '''chances''' of that happening ''(but still considering the result to be precise)''. 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 theorem demonstrating that [[Bayes' theorem]] can be derived from the concept of [[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 [[Dempster-Shafer theory]] and [[rough set]]s. 
 
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;Fuzzy logic will be difficult to scale to larger problems.
 
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:In a widely circulated and highly controversial paper, Charles Elkan in 1993 commented that "...''there are few, if any, published reports of expert systems in real-world use that reason about uncertainty using fuzzy logic. It appears that the limitations of fuzzy logic have not been detrimental in control applications because current fuzzy controllers are far simpler than other knowledge-based systems. In future, the technical limitations of fuzzy logic can be expected to become important in practice, and work on fuzzy controllers will also encounter several problems of scale already known for other knowledge-based systems''". Reactions to Elkan's paper are many and varied, from claims that he is simply mistaken, to others who accept that he has identified important limitations of fuzzy logic that need to be addressed by system designers. In fact, fuzzy logic wasn't largely used at that time, and today it is used to solve very complex problems in the [[Artificial intelligence|AI]] area. Probably the scalability and complexity of the ''fuzzy'' system will depend more on its implementation than on the theory of fuzzy logic.
 
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== Examples where fuzzy logic is used ==
 
  
 
* Automobile and other vehicle subsystems, such as [[Anti-lock braking system|ABS]] and [[cruise control]] (e.g. Tokyo monorail)
 
* Automobile and other vehicle subsystems, such as [[Anti-lock braking system|ABS]] and [[cruise control]] (e.g. Tokyo monorail)
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Fuzzy logic has also been incorporated into some [[microcontroller]]s and [[microprocessor]]s, for instance, the [[Freescale 68HC12]].
 
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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==Formal Fuzzy Logics==
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Fuzzy logic usually uses IF/THEN rules, or constructs that are equivalent, such as [[fuzzy associative matrix|fuzzy associative matrices]].
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Fuzzy logic, when narrowly construed, is an extension of ordinary logics. Here are some examples. The basic idea is that, in fuzzy extensions of logics, formulas can take any values in the unit interval, instead of just 1 or 0 as in ordinary logics.   
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Rules are usually expressed in the form:
 
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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:
 
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IF temperature IS very cold THEN stop fan
 
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IF temperature IS cold THEN turn down fan
 
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IF temperature IS normal THEN maintain level
 
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IF temperature IS hot THEN speed up fan
 
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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 setup a database of "rules" which are queried to deduct logic.  This sort of programming is known as [[logic programming]].
 
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=== Other examples ===
 
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* If a man is 1.8 meters, consider him as tall:
 
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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:
 
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IF height <= medium male THEN is_short IS agree somehow
 
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IF height >= medium male THEN is_tall IS agree somehow
 
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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
 
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small male = (1.3, 1.5]
 
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medium male = (1.5, 1.8]
 
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tall male = (1.8, 2.0]
 
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  giant male > 2.0 m
 
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For the [[consequent]], there are also not only two values, but five, say:
 
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agree not = 0
 
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agree little = 1
 
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agree somehow = 2
 
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agree a lot = 3
 
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agree fully = 4
 
  
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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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=== Basic Fuzzy Propositional Logic ===
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In ''basic fuzzy propositional logic'', formulas are built, as in the language of ordinary [[propositional calculi|propositional logic]], from propositional variables, <math>\rightarrow</math>, <math>\wedge</math> and propositional constant 0. (<math>\lnot \phi</math> is defined as <math>\phi \arrow 0</math>) 
  
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The crisp example differs deliberately from the fuzzy one. We did not put in the [[antecedent]]  
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Interpretation functions on propositional variables are mappings from the set of propositional letters into [0, 1], and truth functional connectives are interpreted in terms of continuous t-norms. A t-norm \triangle is a binary on [0, 1] if \triangle satisfies the following conditions:
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• 1\trianglex=x
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• x\triangle y = y \triangle x
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• x \triangle (y \triangle z) = (x \triangle y) \triangle z
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• if v \leq w and x \leq y, then v \triangle x \leq w \triangle y.
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A binary connective \triangle is continuous if \triangle satisfies:
 +
: for each \epsilon > 0 there is a \delta > 0 such that wherever |x1 -x2| < d and |y1 - y2| < d then |(x1*y1) - (x2*y2)| < e.
 +
Given a t-norm, the residuum \Rightarrow is defined as:
 +
:x \Rightarrow y = max{z| x*z ≤ y}
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A t-norm and its residuum interpret \wedge and \rightarrow, and 0 in [0, 1] interprets the constant 0. Given an interpretation function e on propositional variables, a t-norm induces a valuation function e\triangle on every formula.
 +
A formula \phi is called a t-tautology if e\triangle=1.
  
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IF male >= agree somehow AND ...
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The axiomatic system BL of basic fuzzy propositional logic consists of the following t-tautologies as its axioms and [[Modus Ponens]] as the inferential rules: (A1)-(A7).
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This system is sound and complete in the sense that a formula \phi is a t-tautology if and only if \phi is provable in BL.
  
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as gender is often considered as a binary information. So, it is not so complex like being tall.
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===Versions of Fuzzy Propositional Logic===  
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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]] is an axiomatization of logic where the [[conjunction]] is defined by a continuous [[triangular norm|t-norm]], and the implication defined as 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 axioms of basic logic plus additional axiom of double negation (so it is not [[intuitionistic logic]]), and its models correspond to [[MV-algebra]]s.
 
* [[Lukasiewicz fuzzy logic|Łukasiewicz fuzzy logic]] is a special case of basic fuzzy logic where conjunction is Łukasiewicz t-norm. It has axioms of basic logic plus additional 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 axioms of basic logic plus additional axiom of idempotence of conjunction, and its models are called [[G-algebra]]s.
 
* [[Godel fuzzy logic|Gödel fuzzy logic]] is a special case of basic fuzzy logic where conjunction is Gödel t-norm. It has axioms of basic logic plus additional 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 axioms of basic logic plus additional axiom, and its models are called [[product algebra]]s.
 
* [[Product fuzzy logic]] is a special case of basic fuzzy logic where conjunction is product t-norm. It has axioms of basic logic plus additional axiom, and its models are called [[product algebra]]s.
 
* [[Rational Pavelka logic]] is a generalization of [[multi-valued logic]]. It is an extension of Łukasziewicz fuzzy logic with additional constants.
 
* [[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).
 
  
 
=== Predicate fuzzy logics ===
 
=== 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 how [[predicate logic]] is created from [[propositional logic]].
 
These extend the above-mentioned fuzzy logics by adding [[universal quantifier|universal]] and [[existential quantifier]]s in a manner similar to the way how [[predicate logic]] is created from [[propositional logic]].
 +
 +
== Misconceptions and 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 something cannot be "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]].
 +
 +
;Fuzzy logic is a new way of expressing probability.
 +
:Fuzzy logic and probability refer to different kinds of uncertainty. Fuzzy logic is specifically designed to deal with '''imprecision''' of facts (fuzzy logic statements), while probability deals with '''chances''' of that happening ''(but still considering the result to be precise)''. 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 theorem demonstrating that [[Bayes' theorem]] can be derived from the concept of [[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 [[Dempster-Shafer theory]] and [[rough set]]s. 
 +
 +
;Fuzzy logic will be difficult to scale to larger problems.
 +
:In a widely circulated and highly controversial paper, Charles Elkan in 1993 commented that "...''there are few, if any, published reports of expert systems in real-world use that reason about uncertainty using fuzzy logic. It appears that the limitations of fuzzy logic have not been detrimental in control applications because current fuzzy controllers are far simpler than other knowledge-based systems. In future, the technical limitations of fuzzy logic can be expected to become important in practice, and work on fuzzy controllers will also encounter several problems of scale already known for other knowledge-based systems''". Reactions to Elkan's paper are many and varied, from claims that he is simply mistaken, to others who accept that he has identified important limitations of fuzzy logic that need to be addressed by system designers. In fact, fuzzy logic wasn't largely used at that time, and today it is used to solve very complex problems in the [[Artificial intelligence|AI]] area. Probably the scalability and complexity of the ''fuzzy'' system will depend more on its implementation than on the theory of fuzzy logic.
 +
  
 
== See also ==
 
== See also ==

Revision as of 02:18, 18 August 2006

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Fuzzy logic, when construed in a wider sense, is the theory of fuzzy sets. Given a set U, fuzzy sets are mappings from U into the unit interval [0, 1]. The value that a given fuzzy set A assigns to an element x in U is called the degree of the membership of x in the fuzzy set A, and, in this sense, fuzzy sets are regarded as generalizing the concept of sets in ordinary set theory.

The concept of fuzzy sets provides a convenient way to represent various notions with imprecision, vagueness, or fuzziness, e.g. young, tall, cold, etc., which we frequently employ in our everyday life. As such, fuzzy logic has been applied for various engineering purposes.

When construed in a narrower sense, fuzzy logic is an extension of ordinary two-valued logics in such a way that the points in the unit interval are allowed as truth-values. As the truth-values are generalized in such a way, usual truth-functional operations are generalized accordingly.

Fuzzy logic is controversial in some circles, despite wide acceptance and a broad track record of successful applications. It is rejected by some control engineers for validation and other reasons, and by some 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 sets.

Fuzzy Sets

Fuzzy logic studies fuzzy sets, which was first introduced by L. Zadeh in 1965. Zadeh maintains that the meanings of many words in natural language come with degrees. 12 years old and 18 years old are clearly both young; however 12 years old is younger than 18 years old. To represent this, he introduces the concept of fuzzy subsets. A fuzzy subset of a given set U is a function from U into [1, 0]. The value that a given fuzzy set A assigns to an element x in U is called the degree of the membership of x in the fuzzy set A. Fuzzy subsets are usually referred to simply as fuzzy sets. Using this framework, the meaning of, say, the word “young” can be represented. Take the set of natural numbers and define some fuzzy set, as you like, that assigns values in the unit interval to natural numbers so that, say, 12 (years old) get some value (e.g. .95) higher than the value that 18 (e.g. .85). In that case, the value that each number gets assigned represents the degree of youth. The degree of the membership of 12 in the “youth” subset is higher than that of 18.

This concept of fuzzy sets generalizes the concept of sets in ordinary set theory. Given a set U, a subset S, in the ordinary sense, of U are determined by a function from U to {1, 0}. The elements of U that get 1 assigned represent the elements in S and the elements that get 0 assigned represent the elements that are not in S. The elements of U are all either in, or not in, the subset. However, fuzzy subsets are allowed to take any value in the unit interval other than just 1 and 0. In this sense, the sets in the ordinary sense are special cases of fuzzy sets.

Applications in Engineering

Examples

Fuzzy logic can be used to control household appliances such as washing machines (which sense load size and detergent concentration and adjust their wash cycles accordingly) and refrigerators.

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.

Warm fuzzy logic member function.gif

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. 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".

A more sophisticated practical example is the use of fuzzy logic in high-performance error correction to improve information reception over a limited-bandwidth communication link affected by data-corrupting noise using turbo codes. The front-end of a decoder produces a likelihood measure for the value intended by the sender (0 or 1) for each bit in the data stream. The likelihood measures might use a scale of 256 values between extremes of "certainly 0" and "certainly 1". Two decoders may analyse the data in parallel, arriving at different likelihood results for the values intended by the sender. Each can then use as additional data the other's likelihood results, and repeats the process to improve the results until consensus is reached as to the most likely values.

Other Examples

  • Automobile and other vehicle subsystems, such as ABS and cruise control (e.g. Tokyo monorail)
  • Air conditioners
  • The MASSIVE engine used in the Lord of the Rings films, which helped show huge scale armies create random, yet orderly movements
  • Cameras
  • Digital image processing, such as edge detection
  • Rice cookers
  • Dishwashers
  • Elevators
  • Washing machines and other home appliances
  • Video game artificial intelligence
  • Language filters on message boards and chat rooms for filtering out offensive text
  • Pattern recognition in Remote Sensing

Fuzzy logic has also been incorporated into some microcontrollers and microprocessors, for instance, the Freescale 68HC12.

Formal Fuzzy Logics

Fuzzy logic, when narrowly construed, is an extension of ordinary logics. Here are some examples. The basic idea is that, in fuzzy extensions of logics, formulas can take any values in the unit interval, instead of just 1 or 0 as in ordinary logics.

Basic Fuzzy Propositional Logic

In basic fuzzy propositional logic, formulas are built, as in the language of ordinary propositional logic, from propositional variables, , and propositional constant 0. ( is defined as Failed to parse (unknown function "\arrow"): {\displaystyle \phi \arrow 0} )

Interpretation functions on propositional variables are mappings from the set of propositional letters into [0, 1], and truth functional connectives are interpreted in terms of continuous t-norms. A t-norm \triangle is a binary on [0, 1] if \triangle satisfies the following conditions: • 1\trianglex=x • x\triangle y = y \triangle x • x \triangle (y \triangle z) = (x \triangle y) \triangle z • if v \leq w and x \leq y, then v \triangle x \leq w \triangle y. A binary connective \triangle is continuous if \triangle satisfies:

for each \epsilon > 0 there is a \delta > 0 such that wherever |x1 -x2| < d and |y1 - y2| < d then |(x1*y1) - (x2*y2)| < e.

Given a t-norm, the residuum \Rightarrow is defined as:

x \Rightarrow y = max{z| x*z ≤ y}

A t-norm and its residuum interpret \wedge and \rightarrow, and 0 in [0, 1] interprets the constant 0. Given an interpretation function e on propositional variables, a t-norm induces a valuation function e\triangle on every formula. A formula \phi is called a t-tautology if e\triangle=1.

The axiomatic system BL of basic fuzzy propositional logic consists of the following t-tautologies as its axioms and Modus Ponens as the inferential rules: (A1)-(A7). This system is sound and complete in the sense that a formula \phi is a t-tautology if and only if \phi is provable in BL.

Versions of Fuzzy Propositional Logic

  • Łukasiewicz fuzzy logic is a special case of basic fuzzy logic where conjunction is Łukasiewicz t-norm. It has axioms of basic logic plus additional 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 axioms of basic logic plus additional 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 axioms of basic logic plus additional axiom, and its models are called product algebras.
  • Rational Pavelka logic is a generalization of multi-valued logic. It is an extension of Łukasziewicz fuzzy logic with additional constants.

Predicate fuzzy logics

These extend the above-mentioned fuzzy logics by adding universal and existential quantifiers in a manner similar to the way how predicate logic is created from propositional logic.

Misconceptions and 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 something cannot be "cold" at N degrees but "not cold" at N+1 degrees — a concept classical logic cannot easily handle due to the principle of bivalence.
Fuzzy logic is a new way of expressing probability.
Fuzzy logic and probability refer to different kinds of uncertainty. Fuzzy logic is specifically designed to deal with imprecision of facts (fuzzy logic statements), while probability deals with chances of that happening (but still considering the result to be precise). 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 theorem demonstrating that Bayes' theorem can be derived from the concept of 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 Dempster-Shafer theory and rough sets.
Fuzzy logic will be difficult to scale to larger problems.
In a widely circulated and highly controversial paper, Charles Elkan in 1993 commented that "...there are few, if any, published reports of expert systems in real-world use that reason about uncertainty using fuzzy logic. It appears that the limitations of fuzzy logic have not been detrimental in control applications because current fuzzy controllers are far simpler than other knowledge-based systems. In future, the technical limitations of fuzzy logic can be expected to become important in practice, and work on fuzzy controllers will also encounter several problems of scale already known for other knowledge-based systems". Reactions to Elkan's paper are many and varied, from claims that he is simply mistaken, to others who accept that he has identified important limitations of fuzzy logic that need to be addressed by system designers. In fact, fuzzy logic wasn't largely used at that time, and today it is used to solve very complex problems in the AI area. Probably the scalability and complexity of the fuzzy system will depend more on its implementation than on the theory of fuzzy logic.


See also

  • Artificial intelligence
  • Artificial neural network
  • Neuro-fuzzy
  • Biologically-inspired computing
  • Combs method
  • Control system
  • Defuzzification
  • Dynamic logic
  • Expert system
  • Fuzzy associative matrix
  • Fuzzy concept
  • Fuzzy Control Language
  • Fuzzy control system
  • Fuzzy electronics
  • Fuzzy set
  • Machine learning
  • Paradox of the heap
  • Pattern recognition
  • Rough set
  • FSQL (Fuzzy SQL, a fuzzy language for databases)

Bibliography

  • Constantin von Altrock, Fuzzy Logic and NeuroFuzzy Applications Explained (2002), ISBN 0-13-368465-2
  • Earl Cox, The Fuzzy Systems Handbook (1994), ISBN 0-12-194270-8
  • Charles Elkan. The Paradoxical Success of Fuzzy Logic. November 1993. Available from Elkan's home page.
  • Petr Hájek: Metamathematics of fuzzy logic. Kluwer 1998.
  • Frank Höppner, Frank Klawonn, Rudolf Kruse and Thomas Runkler, Fuzzy Cluster Analysis (1999), ISBN 0-471-98864-2
  • George Klir and Tina Folger, Fuzzy Sets, Uncertainty, and Information (1988), ISBN 0-13-345984-5
  • George Klir, UTE H. St.Clair and Bo Yuan Fuzzy Set Theory Foundations and Applications,1997,
  • George Klir and Bo Yuan, Fuzzy Sets and Fuzzy Logic (1995) ISBN 0-13-101171-5
  • Bart Kosko, Fuzzy Thinking: The New Science of Fuzzy Logic (1993), Hyperion. ISBN 078688021X
  • Ronald Yager and Dimitar Filev, Essentials of Fuzzy Modeling and Control (1994), ISBN 0-471-01761-2
  • Hans-Jürgen Zimmermann, Fuzzy Set Theory and its Applications (2001), ISBN 0792374355
  • Kevin M. Passino and Stephen Yurkovich, Fuzzy Control, Addison Wesley Longman, Menlo Park, CA, 1998.

External links

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