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Encyclopedia > Fuzzy logic

Fuzzy logic is derived from fuzzy set theory dealing with reasoning that is approximate rather than precisely deduced from classical 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). Image File history File links This is a lossless scalable vector image. ... Fuzzy Logic is the name of the debut album by the Super Furry Animals. ... Fuzzy sets are an extension of classical set theory and are used in fuzzy logic. ... Reasoning is the mental (cognitive) process of looking for reasons to support beliefs, conclusions, actions or feelings. ... Propositional Logic (PL) is a system for evaluating the validity of arguments by encoding them into sentential variables and boolean operator and is part of the philosophy of Formal logic, //  Explanation Note that the actual truth of the premises are not particularly relevant in PL; it is dealing mostly...

Degrees of truth are often confused with probabilities. However, they are distinct conceptually; fuzzy truth represents membership in vaguely defined sets, not likelihood of some event or condition. 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'. Probability is the likelihood that something is the case or will happen. ... The membership function of a fuzzy set corresponds to the indicator function of classical sets. ... A scalar may be: Look up scalar in Wiktionary, the free dictionary. ...

Fuzzy logic allows for set membership values to range (inclusively) between 0 and 1, 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 Lotfi Zadeh at the University of California, Berkeley. The membership function of a fuzzy set corresponds to the indicator function of classical sets. ... Fuzzy sets are an extension of the classical set theory used in Fuzzy logic. ... Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. ... Year 1965 (MCMLXV) was a common year starting on Friday (link will display full calendar) of the 1965 Gregorian calendar. ... Lotfali Askar Zadeh (born February 4, 1921) is a mathematician and computer scientist, and a professor of computer science at the University of California, Berkeley. ... Sather Tower (the Campanile) looking out over the San Francisco Bay and Mount Tamalpais. ...

Fuzzy logic is controversial in some circles and is rejected by some control engineers and by most statisticians who hold that probability is the only rigorous mathematical description of uncertainty.[verification needed] Critics also argue that it cannot be a superset of ordinary set theory since membership functions are defined in terms of conventional sets. For control theory in psychology and sociology, see control theory (sociology). ... This article is about the field of statistics. ... Probability is the likelihood that something is the case or will happen. ... â€œUncertainâ€ redirects here. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... The notion of a set is one of the most important and fundamental concepts in modern mathematics. ...

## Applications

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 control system is a device or set of devices to manage, command, direct or regulate the behaviour of other devices or systems. ... A major appliance is a large machine which accomplishes some routine housekeeping task, which includes purposes such as cooking, food preservation, or cleaning, whether in a household, institutional, commercial or industrial setting. ... Front-loading washing machine. ... Laundry detergents are just one of many possible uses for detergents Detergent is a compound, or a mixture of compounds, intended to assist cleaning. ... Fridge redirects here. ...

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. For the particular temperature illustrated with the vertical line, the three truth values (0.8, 0.2, and 0) could be interpreted as evaluating that particular temperature as being, say, "fairly cold" (blue arrow), "slightly warm" (yellow arrow), and "not hot" (red arrow). In logic and mathematics, a logical value, also called a truth value, is a value indicating to what extent a proposition is true. ...

## 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 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, 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.
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.

In logic, the principle of bivalence states that for any proposition P, either P is true or P is false. ... Bayesianism is the philosophical tenet that the mathematical theory of probability applies to the degree of plausibility of statements, or to the degree of belief of rational agents in the truth of statements; when used with Bayes theorem, it then becomes Bayesian inference. ... Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. ... In mathematics, a probability space is a set S, together with a σ-algebra X on S and a measure P on that σ-algebra such that P(S) = 1. ... Statisticians or people who made notable contributions to the theories of statistics, or related aspects of probability, or machine learning: Peter Armitage M. S. Bartlett Thomas Bayes Yves Berger Duane Boes Ladislaus Bortkiewicz George Box Pafnuty Chebyshev Alexey Chervonenkis William Cochran (Sir) David R. Cox Richard Threlkeld Cox Harald Cram... Bruno de Finetti (Innsbruck, June 13, 1906 - Rome, July 20, 1985) was an Italian probabilist and statistician, noted for the operational subjective conception of probability. ... Bart Kosko is professor of electrical engineering at the University of Southern California (USC). ... In probability theory, Bayes theorem (often called Bayes Law) relates the conditional and marginal probabilities of two random events. ... Fuzzy sets are an extension of classical set theory and are used in fuzzy logic. ... Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. ... The Dempster-Shafer theory is a mathematical theory of evidence [SH76] based on belief functions and plausible reasoning, which is used to combine separate pieces of information (evidence) to calculate the probability of an event. ... Cover of Rough Sets: Theoretical Aspects of Reasoning about Data by ZdzisÅ‚aw Pawlak (Kluwer 1991). ...

## How fuzzy logic is applied

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. A fuzzy associative matrix expresses fuzzy logic rules in matrix form. ...

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: In logical calculus, logical operators or logical connectors serve to connect statements into more complicated compound statements. ... Boolean logic is a complete system for logical operations. ...

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. A programming language is an artificial language that can be used to control the behavior of a machine, particularly a computer. ... Prolog is a logic programming language. ... Logic programming (which might better be called logical programming by analogy with mathematical programming and linear programming) is, in its broadest sense, the use of mathematical logic for computer 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. A relational database is a database that conforms to the relational model, and refers to a databases data and schema (the databases structure of how those data are arranged). ... Maria Zemankova Maria Zemankova is a Computer Scientist who is known for the theory and implementation of the first Fuzzy Relational Database System. ... This article is about the city in Saudi Arabia. ... For the Slavic nymphs of this name, see Slavic fairies. ... Bosc may refer to a type of pear, the Bosc pear. ... SQL (IPA: or ) is a computer language designed for the retrieval and management of data in relational database management systems, database schema creation and modification, and database object access control management. ...

### 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:
A consequent is the second half of a hypothetical proposition. ...

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 An antecedent is a preceding phrase or word. ...

IF male >= agree somewhat AND ...

as gender is often considered as a binary information. So, it is not so complex as being tall.

## 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. Mathematical logic is a major area of mathematics, which grew out of symbolic logic. ... In logic and mathematics, a formal system consists of two components, a formal language plus a set of inference rules or transformation rules. ...

### 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). In mathematics, a T-norm (or t-norm) is a kind of binary operation used in the framework of probabilistic metric spaces and in multi-valued logic, specifically in fuzzy logic. ... In the mathematical discipline of model theory, a structure for a language (referred to as an -structure, and commonly written as a Gothic capital) is an ordered pair whose first member is the domain of discourse or universe set (taken to be a set with possibly relations and functions defined... Lukasiewicz logic is a real-valued logic in which sentences from sentential calculus are assigned a truth value between 0 and 1. ... Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ... This article may be too technical for most readers to understand, and needs attention from an expert on its subject. ... Kurt GÃ¶del Kurt GÃ¶del [kurt gÃ¸Ëdl], (April 28, 1906 â€“ January 14, 1978) was a logician, mathematician, and philosopher of mathematics. ... Monoidal t-norm based logic (or shortly MTL), the logic of left-continuous t-norms, is a formal system of propositional fuzzy logic. ... In modern algebra, a residuated lattice is a lattice with certain simple properties which apply to collections of all two-sided ideals of any ring. ... Multi-valued logics are logical calculi in which there are more than two possible truth values. ... Propositional logic or sentential logic is the logic of propositions, sentences, or clauses. ... In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ...

### 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. In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ... In predicate logic, existential quantification is an attempt to formalize the notion that something (a logical predicate) is true for something, or at least one relevant thing. ... ... Propositional logic or sentential logic is the logic of propositions, sentences, or clauses. ... In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is less than or equal to all other elements of the subset. ... In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ...

### Effectiveness for fuzzy logics

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 $rightarrow$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). 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). In computability theory, often less suggestively called recursion theory, a countable set S is called recursively enumerable, computably enumerable, semi-decidable or provable if There is an algorithm that, when given an input — typically an integer or a tuple of integers or a sequence of characters — eventually halts if it... Classical mathematics, as a term of art in mathematical logic, refers generally to mathematics constructed and proved on the basis of classical logic and ZFC set theory, i. ... Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. ... For the test of artificial intelligence, see Turing test. ...

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. Kurt GÃ¶del Kurt GÃ¶del [kurt gÃ¸Ëdl], (April 28, 1906 â€“ January 14, 1978) was a logician, mathematician, and philosopher of mathematics. ...

## Bibliography

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• 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.
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• Wiedermann, J. (2004). "Characterizing the super-Turing computing power and efficiency of classical fuzzy Turing machines". Theor. Comput. Sci. 317: 61-69.
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• Zimmermann, H. (2001). Fuzzy set theory and its applications. Boston: Kluwer Academic Publishers. ISBN 0-7923-7435-5.

A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... ISSN, or International Standard Serial Number, is the unique eight-digit number applied to a periodical publication including electronic serials. ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... ISSN, or International Standard Serial Number, is the unique eight-digit number applied to a periodical publication including electronic serials. ... ISSN, or International Standard Serial Number, is the unique eight-digit number applied to a periodical publication including electronic serials. ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... ISSN, or International Standard Serial Number, is the unique eight-digit number applied to a periodical publication including electronic serials. ... George Jiri Klir (1932 Prague, Czechoslovakia) is an Czech-American computer scientist and professor of systems sciences at the Center for Intelligent Systems at the Binghamton University in New York. ... Bart Kosko is professor of electrical engineering at the University of Southern California (USC). ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... ISSN, or International Standard Serial Number, is the unique eight-digit number applied to a periodical publication including electronic serials. ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... ISSN, or International Standard Serial Number, is the unique eight-digit number applied to a periodical publication including electronic serials. ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... ISSN, or International Standard Serial Number, is the unique eight-digit number applied to a periodical publication including electronic serials. ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... ISSN, or International Standard Serial Number, is the unique eight-digit number applied to a periodical publication including electronic serials. ...

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 Fuzzy logic tutorial. (3115 words) Fuzzy logic is used in system control and analysis design, because it shortens the time for engineering development and sometimes, in the case of highly complex systems, is the only way to solve the problem. Fuzzy logic control and analysis systems may be electro-mechanical in nature, or concerned only with data, for example economic data, in all cases guided by "If-Then rules" stated in human language. It is based on fuzzy perceptions, fuzzy truths, fuzzy inferences, etc., all resulting in an averaged, summarized, normalized output, which is given by the human a precise number or decision value which he or she verbalizes, writes down or acts on.
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