# Fuzzy logic

Fuzzy logic, when construed in a wider sense, is the theory of fuzzy sets. The concept of fuzzy sets provides a convenient way to represent various notions with imprecision, vagueness, or fuzziness, for example young, tall, cold, and so forth, which we frequently employ in our everyday life. As such, fuzzy logic has the rationale of more closely resembling than traditional logic the way human beings actually think, where alternatives are not black and white but shades of gray. Fuzzy logic has had notable success in various engineering applications.

When construed in a narrower sense, fuzzy logic is an extension of ordinary two-valued logic in such a way that the points in interval units 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. Twelve 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.

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 analyze 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. 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, truth-functional connectives, $\rightarrow$, $\wedge$, and propositional constant 0. ($\lnot \phi$ is defined as $\phi \rightarrow 0$)

Interpretation functions on propositional variables are mappings from the set of propositional variables into [0, 1], and truth functional connectives are interpreted in terms of continuous t-norms. A t-norm $\triangle$ is a binary operator on [0, 1] if $\triangle$ satisfies the following conditions:

• $1 \triangle x=x$
• $x\triangle y = y \triangle x$
• $x \triangle (y \triangle z) = (x \triangle y) \triangle z$
• 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 $|x_1 -x_2| < \delta$ and $|y_1 - y_2| < \delta$, then $|(x_1 \triangle y_1) - (x_2 \triangle y_2)| < \epsilon$.

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

$x \Rightarrow y$ = max { z| $x \triangle z \leq y$}

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 $e_\triangle=1$.

There is a sound and complete axiomatization, i.e. the system in which a formula $\phi$is a t-tautology if and only if $\phi$ is provable.

### Versions of Fuzzy Propositional Logic

• _ukasiewicz fuzzy logic is a special case of basic fuzzy logic where conjunction is _ukasiewicz 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 _ukasziewicz fuzzy logic with additional constants.

### Basic Fuzzy Predicate Logic

The language of basic fuzzy predicate logic consists of the same items as the first-order logic (variables, predicate symbols, $\wedge$, $\rightarrow$, 0, quantifiers). An interpretation consists of a nonempty domain and a function that maps an n-ary predicate symbol to an n-ary fuzzy relation (an n-ary fuzzy relation here is a mapping from n-ary tuples of objects in the domain to values in [0, 1]. An n-ary fuzzy relation that corresponds to a predicate symbol R represents the degrees in which n-ary tuples satisfy the formula $Rx_1...x_n$. Given a continuous t-norm, the connectives are interpreted as in the case of basic fuzzy propositional logic. The truth degree of a formula of the form $\forall x \phi$ is defined as the infimum of the truth degrees of the instances of $\phi$, and that of a formula of the form $\exists x \phi$ is defined as the supremum of the instances of $\phi$. The interpretations of basic fuzzy predicate logic generalize to so-called BL-algebra, and, based on the interpretation, a sound and complete axiomatization can be given (see Hájek 1998 for details).

## 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 in 1993, Charles Elkan 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.

## Bibliography

• von Altrock, Constantin. 2002. Fuzzy Logic and NeuroFuzzy Applications Explained. ISBN 0133684652
• Cox, Earl. 1994. The Fuzzy Systems Handbook. ISBN 0121942708
• Elkan, Charles. The Paradoxical Success of Fuzzy Logic. November 1993. Available from Elkan's home page Retrieved September 18, 2008.
• Hájek, Petr. “Metamathematics of fuzzy logic.” Kluwer, 1998.
• Höppner, Frank, Frank Klawonn, Rudolf Kruse and Thomas Runkler. 1999. Fuzzy Cluster Analysis. ISBN 0471988642
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• Kosko, Bart. 1993. Fuzzy Thinking: The New Science of Fuzzy Logic. Hyperion. ISBN 078688021X
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• Zimmermann, Hans-Jürgen. 2001. Fuzzy Set Theory and its Applications. ISBN 0792374355