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Fuzzy logic is a form of
many-valued logic Many-valued logic (also multi- or multiple-valued logic) refers to a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values (i.e., "true" and "false ...
in which the
truth value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false''). Computing In some pro ...
of variables may be any
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely false. By contrast, in Boolean logic, the truth values of variables may only be the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
values 0 or 1. The term ''fuzzy logic'' was introduced with the 1965 proposal of
fuzzy set theory In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, defined ...
by Iranian Azerbaijani mathematician
Lotfi Zadeh Lotfi Aliasker Zadeh (; az, Lütfi Rəhim oğlu Ələsgərzadə; fa, لطفی علی‌عسکرزاده; 4 February 1921 – 6 September 2017) was a mathematician, computer scientist, electrical engineer, artificial intelligence researcher, an ...
. Fuzzy logic had, however, been studied since the 1920s, as infinite-valued logic—notably by Łukasiewicz and Tarski. Fuzzy logic is based on the observation that people make decisions based on imprecise and non-numerical information. Fuzzy models or sets are mathematical means of representing
vagueness In linguistics and philosophy, a vague predicate is one which gives rise to borderline cases. For example, the English adjective "tall" is vague since it is not clearly true or false for someone of middling height. By contrast, the word "prime" is ...
and imprecise information (hence the term fuzzy). These models have the capability of recognising, representing, manipulating, interpreting, and using data and information that are vague and lack certainty. Fuzzy logic has been applied to many fields, from
control theory Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
to
artificial intelligence Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speech r ...
.

# Overview

Classical logic only permits conclusions that are either true or false. However, there are also propositions with variable answers, such as one might find when asking a group of people to identify a color. In such instances, the truth appears as the result of reasoning from inexact or partial knowledge in which the sampled answers are mapped on a spectrum. Both
degrees of truth In classical logic, propositions are typically unambiguously considered as being true or false. For instance, the proposition ''one is both equal and not equal to itself'' is regarded as simply false, being contrary to the Law of Noncontradiction; ...
and probabilities range between 0 and 1 and hence may seem similar at first, but fuzzy logic uses degrees of truth as a mathematical model of ''vagueness'', while
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
is a mathematical model of ''ignorance''.

## Applying truth values

A basic application might characterize various sub-ranges of a continuous variable. For instance, a temperature measurement for
anti-lock brakes An anti-lock braking system (ABS) is a automobile safety, safety anti-Skid (automobile), skid Brake, braking system used on aircraft and on land motor vehicle, vehicles, such as cars, motorcycles, trucks, and buses. ABS operates by preventing t ...
might have several separate membership functions defining particular temperature ranges needed to control the brakes properly. Each function maps the same temperature value to a truth value in the 0 to 1 range. These truth values can then be used to determine how the brakes should be controlled. Fuzzy set theory provides a means for representing uncertainty.

## Linguistic variables

In fuzzy logic applications, non-numeric values are often used to facilitate the expression of rules and facts. A linguistic variable such as ''age'' may accept values such as ''young'' and its antonym ''old''. Because natural languages do not always contain enough value terms to express a fuzzy value scale, it is common practice to modify linguistic values with
adjectives In linguistics, an adjective (abbreviated ) is a word that generally modifies a noun or noun phrase or describes its referent. Its semantic role is to change information given by the noun. Traditionally, adjectives were considered one of the mai ...
or
adverbs An adverb is a word or an expression that generally modifies a verb, adjective, another adverb, determiner, clause, preposition, or sentence. Adverbs typically express manner, place, time, frequency, degree, level of certainty, etc., answering que ...
. For example, we can use the hedges ''rather'' and ''somewhat'' to construct the additional values ''rather old'' or ''somewhat young''.

# Fuzzy systems

## Mamdani

The most well-known system is the Mamdani rule-based one. It uses the following rules: # Fuzzify all input values into fuzzy membership functions. # Execute all applicable rules in the rulebase to compute the fuzzy output functions. # De-fuzzify the fuzzy output functions to get "crisp" output values.

### Fuzzification

Fuzzification is the process of assigning the numerical input of a system to fuzzy sets with some degree of membership. This degree of membership may be anywhere within the interval ,1 If it is 0 then the value does not belong to the given fuzzy set, and if it is 1 then the value completely belongs within the fuzzy set. Any value between 0 and 1 represents the degree of uncertainty that the value belongs in the set. These fuzzy sets are typically described by words, and so by assigning the system input to fuzzy sets, we can reason with it in a linguistically natural manner. For example, in the image below the meanings of the expressions ''cold'', ''warm'', and ''hot'' are represented by functions mapping a temperature scale. A point on that scale has three "truth values"—one for each of the three functions. The vertical line in the image represents a particular temperature that the three arrows (truth values) gauge. Since the red arrow points to zero, this temperature may be interpreted as "not hot"; i.e. this temperature has zero membership in the fuzzy set "hot". The orange arrow (pointing at 0.2) may describe it as "slightly warm" and the blue arrow (pointing at 0.8) "fairly cold". Therefore, this temperature has 0.2 membership in the fuzzy set "warm" and 0.8 membership in the fuzzy set "cold". The degree of membership assigned for each fuzzy set is the result of fuzzification.
Fuzzy set In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, defined ...
s are often defined as triangle or trapezoid-shaped curves, as each value will have a slope where the value is increasing, a peak where the value is equal to 1 (which can have a length of 0 or greater) and a slope where the value is decreasing. They can also be defined using a
sigmoid function A sigmoid function is a mathematical function having a characteristic "S"-shaped curve or sigmoid curve. A common example of a sigmoid function is the logistic function shown in the first figure and defined by the formula: :S(x) = \frac = \ ...
. One common case is the standard logistic function defined as : $S\left(x\right) = \frac$ which has the following symmetry property : $S(x) + S(-x) = 1$ From this it follows that $(S(x) + S(-x)) \cdot (S(y) + S(-y)) \cdot (S(z) + S(-z)) = 1$

### Fuzzy logic operators

Fuzzy logic works with membership values in a way that mimics Boolean logic. To this end, replacements for basic operators AND, OR, NOT must be available. There are several ways to this. A common replacement is called the ''s'': For TRUE/1 and FALSE/0, the fuzzy expressions produce the same result as the Boolean expressions. 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 science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ...
. However, an arbitrary choice table does not always define a fuzzy logic function. In the paper (Zaitsev, et al), a criterion has been formulated to recognize whether a given choice table defines a fuzzy logic function and a simple algorithm of fuzzy logic function synthesis has been proposed based on introduced concepts of constituents of minimum and maximum. A fuzzy logic function represents a disjunction of constituents of minimum, where a constituent of minimum is a conjunction of variables of the current area greater than or equal to the function value in this area (to the right of the function value in the inequality, including the function value). Another set of AND/OR operators is based on multiplication, where x AND y = x*y NOT x = 1 - x Hence, x OR y = NOT( AND( NOT(x), NOT(y) ) ) x OR y = NOT( AND(1-x, 1-y) ) x OR y = NOT( (1-x)*(1-y) ) x OR y = 1-(1-x)*(1-y) x OR y = x+y-xy Given any two of AND/OR/NOT, it is possible to derive the third. The generalization of AND is known as a
t-norm In mathematics, a t-norm (also T-norm or, unabbreviated, triangular norm) is a kind of binary operation used in the framework of probabilistic metric spaces and in multi-valued logic, specifically in fuzzy logic. A t-norm generalizes intersectio ...
.

### IF-THEN rules

IF-THEN rules map input or computed truth values to desired output truth values. Example: IF temperature IS very cold THEN fan_speed is stopped IF temperature IS cold THEN fan_speed is slow IF temperature IS warm THEN fan_speed is moderate IF temperature IS hot THEN fan_speed is high Given a certain temperature, the fuzzy variable ''hot'' has a certain truth value, which is copied to the ''high'' variable. Should an output variable occur in several THEN parts, then the values from the respective IF parts are combined using the OR operator.

### Defuzzification

The goal is to get a continuous variable from fuzzy truth values. This would be easy if the output truth values were exactly those obtained from fuzzification of a given number. Since, however, all output truth values are computed independently, in most cases they do not represent such a set of numbers. One has then to decide for a number that matches best the "intention" encoded in the truth value. For example, for several truth values of fan_speed, an actual speed must be found that best fits the computed truth values of the variables 'slow', 'moderate' and so on. There is no single algorithm for this purpose. A common algorithm is # For each truth value, cut the membership function at this value # Combine the resulting curves using the OR operator # Find the center-of-weight of the area under the curve # The x position of this center is then the final output.

## Takagi–Sugeno–Kang (TSK)

The TSK system is similar to Mamdani, but the defuzzification process is included in the execution of the fuzzy rules. These are also adapted, so that instead the consequent of the rule is represented through a polynomial function (usually constant or linear). An example of a rule with a constant output would be: IF temperature IS very cold = 2 In this case, the output will be equal to the constant of the consequent (e.g. 2). In most scenarios we would have an entire rule base, with 2 or more rules. If this is the case, the output of the entire rule base will be the average of the consequent of each rule i (Yi), weighted according to the membership value of its antecedent (hi): $\frac$ An example of a rule with a linear output would be instead: IF temperature IS very cold AND humidity IS high = 2 * temperature + 1 * humidity In this case, the output of the rule will be the result of function in the consequent. The variables within the function represent the membership values after fuzzification, not the crisp values. Same as before, in case we have an entire rule base with 2 or more rules, the total output will be the weighted average between the output of each rule. The main advantage of using TSK over Mamdani is that it is computationally efficient and works well within other algorithms, such as PID control and with optimization algorithms. It can also guarantee the continuity of the output surface. However, Mamdani is more intuitive and easier to work with by people. Hence, TSK is usually used within other complex methods, such as in
adaptive neuro fuzzy inference system An adaptive neuro-fuzzy inference system or adaptive network-based fuzzy inference system (ANFIS) is a kind of artificial neural network that is based on Takagi–Sugeno fuzzy inference system. The technique was developed in the early 1990s. Since ...
s.

# Forming a consensus of inputs and fuzzy rules

Since the fuzzy system output is a consensus of all of the inputs and all of the rules, fuzzy logic systems can be well behaved when input values are not available or are not trustworthy. Weightings can be optionally added to each rule in the rulebase and weightings can be used to regulate the degree to which a rule affects the output values. These rule weightings can be based upon the priority, reliability or consistency of each rule. These rule weightings may be static or can be changed dynamically, even based upon the output from other rules.

# Applications

Fuzzy logic is used in control systems to allow experts to contribute vague rules such as "if you are close to the destination station and moving fast, increase the train's brake pressure"; these vague rules can then be numerically refined within the system. Many of the early successful applications of fuzzy logic were implemented in Japan. A first notable application was on the
Sendai Subway 1000 series The is a rapid transit electric multiple unit (EMU) train type operated on the Sendai Subway Namboku Line in Sendai, Japan. The 1000 series was the world's first train type to use fuzzy logic to control its speed, and this system developed by ...
, in which fuzzy logic was able to improve the economy, comfort, and precision of the ride. It has also been used for
handwriting recognition Handwriting recognition (HWR), also known as handwritten text recognition (HTR), is the ability of a computer to receive and interpret intelligible handwritten input from sources such as paper documents, photographs, touch-screens and other de ...
in Sony pocket computers, helicopter flight aids, subway system controls, improving automobile fuel efficiency, single-button washing machine controls, automatic power controls in vacuum cleaners, and early recognition of earthquakes through the Institute of Seismology Bureau of Meteorology, Japan.

## Artificial intelligence

AI and fuzzy logic, when analyzed, are the same thing — the underlying logic of neural networks is fuzzy. A neural network will take a variety of valued inputs, give them different weights in relation to each other, and arrive at a decision which normally also has a value. Nowhere in that process is there anything like the sequences of either-or decisions which characterize non-fuzzy mathematics, almost all of computer programming, and digital electronics. In the 1980s, researchers were divided about the most effective approach to machine learning: "common sense" models or neural networks. The former approach requires large decision trees and uses binary logic, matching the hardware on which it runs. The physical devices might be limited to binary logic, but AI can use software for its calculations. Neural networks take this approach, which results in more accurate models of complex situations. Neural networks soon found their way onto a multitude of electronic devices.

## Medical decision making

Fuzzy logic is an important concept in medical decision making. Since medical and healthcare data can be subjective or fuzzy, applications in this domain have a great potential to benefit a lot by using fuzzy logic based approaches. Fuzzy logic can be used in many different aspects within the medical decision making framework. Such aspects include in medical image analysis, biomedical signal analysis, segmentation of images or signals, and
feature extraction In machine learning, pattern recognition, and image processing, feature extraction starts from an initial set of measured data and builds derived values (features) intended to be informative and non-redundant, facilitating the subsequent learning a ...
/ selection of images or signals. The biggest question in this application area is how much useful information can be derived when using fuzzy logic. A major challenge is how to derive the required fuzzy data. This is even more challenging when one has to elicit such data from humans (usually, patients). As has been said How to elicit fuzzy data, and how to validate the accuracy of the data is still an ongoing effort strongly related to the application of fuzzy logic. The problem of assessing the quality of fuzzy data is a difficult one. This is why fuzzy logic is a highly promising possibility within the medical decision making application area but still requires more research to achieve its full potential. Although the concept of using fuzzy logic in medical decision making is exciting, there are still several challenges that fuzzy approaches face within the medical decision making framework.

### Image-based computer-aided diagnosis

One of the common application areas that use fuzzy logic is image-based
computer-aided diagnosis Computer-aided detection (CADe), also called computer-aided diagnosis (CADx), are systems that assist doctors in the interpretation of medical images. Imaging techniques in X-ray, MRI, Endoscopy, and ultrasound diagnostics yield a great deal o ...
(CAD) in medicine. CAD is a computerized set of inter-related tools that can be used to aid physicians in their diagnostic decision-making. For example, when a physician finds a lesion that is abnormal but still at a very early stage of development he/she may use a CAD approach to characterize the lesion and diagnose its nature. Fuzzy logic can be highly appropriate to describe key characteristics of this lesion.

## Fuzzy Databases

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 (1983). Later, some other models arose like the Buckles-Petry model, the Prade-Testemale Model, the Umano-Fukami model or the GEFRED model by J. M. Medina, M. A. Vila et al. Fuzzy querying languages have been defined, such as 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 etc.

# Logical analysis

In
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
, there are several
formal system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A form ...
s of "fuzzy logic", most of which are in the family of t-norm fuzzy logics.

## Propositional fuzzy logics

The most important propositional fuzzy logics are: * Monoidal t-norm-based propositional fuzzy logic MTL is an axiomatization of logic where
conjunction Conjunction may refer to: * Conjunction (grammar), a part of speech * Logical conjunction, a mathematical operator ** Conjunction introduction, a rule of inference of propositional logic * Conjunction (astronomy), in which two astronomical bodies ...
is defined by a
left continuous In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
t-norm In mathematics, a t-norm (also T-norm or, unabbreviated, triangular norm) is a kind of binary operation used in the framework of probabilistic metric spaces and in multi-valued logic, specifically in fuzzy logic. A t-norm generalizes intersectio ...
and implication is defined as the residuum of the t-norm. Its
model A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...
s correspond to MTL-algebras that are pre-linear commutative bounded integral
residuated lattice In abstract algebra, a residuated lattice is an algebraic structure that is simultaneously a lattice ''x'' ≤ ''y'' and a monoid ''x''•''y'' which admits operations ''x''\''z'' and ''z''/''y'', loosely analogous to division or implication, when ' ...
s. * Basic propositional fuzzy logic BL is an extension of MTL logic where conjunction is defined by a continuous t-norm, and implication is also defined as the residuum of the t-norm. Its models correspond to BL-algebras. *
Łukasiewicz fuzzy logic Łukasiewicz is a Polish surname. It comes from the given name Łukasz (Lucas). It is found across Poland, particularly in central regions. It is related to the surnames Łukaszewicz and Lukashevich. People * Antoni Łukasiewicz (born 1983 ...
is the extension of basic fuzzy logic BL where standard conjunction is the Łukasiewicz t-norm. It has the axioms of basic fuzzy logic plus an axiom of double negation, and its models correspond to MV-algebras. * Gödel fuzzy logic is the extension of basic fuzzy logic BL where conjunction is the Gödel t-norm (that is, minimum). It has the axioms of BL plus an axiom of idempotence of conjunction, and its models are called G-algebras. * Product fuzzy logic is the extension of basic fuzzy logic BL where conjunction is the product t-norm. It has the axioms of BL plus another axiom for cancellativity of conjunction, and its models are called product algebras. * Fuzzy logic with evaluated syntax (sometimes also called Pavelka's logic), denoted by EVŁ, is a further generalization of mathematical fuzzy logic. While the above kinds of fuzzy logic have traditional syntax and many-valued semantics, in EVŁ syntax is also evaluated. This means that each formula has an evaluation. Axiomatization of EVŁ stems from Łukasziewicz fuzzy logic. A generalization of the classical Gödel completeness theorem is provable in EVŁ.

## Predicate fuzzy logics

Similar to the way
predicate logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
is created from
propositional logic Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
, predicate fuzzy logics extend fuzzy systems by
universal Universal is the adjective for universe. Universal may also refer to: Companies * NBCUniversal, a media and entertainment company ** Universal Animation Studios, an American Animation studio, and a subsidiary of NBCUniversal ** Universal TV, a ...
and
existential quantifier In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, w ...
s. The semantics of the universal quantifier in t-norm fuzzy logics is the
infimum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
of the truth degrees of the instances of the quantified subformula while the semantics of the existential quantifier is the supremum of the same.

## Decidability Issues

The notions of a "decidable subset" and "
recursively enumerable In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: *There is an algorithm such that the ...
subset" are basic ones for classical mathematics and classical logic. Thus the question of a suitable extension of them to
fuzzy set theory In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, defined ...
is a crucial one. The first proposal in such a direction was made by E. S. Santos by the notions of ''fuzzy
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...
'', ''Markov normal fuzzy algorithm'' and ''fuzzy program'' (see Santos 1970). Successively, L. Biacino and G. Gerla argued that the proposed definitions are rather questionable. For example, in one shows that the fuzzy Turing machines are not adequate for fuzzy language theory since there are natural fuzzy languages intuitively computable that cannot be recognized by a fuzzy Turing Machine. Then they proposed the following definitions. Denote by ''Ü'' the set of rational numbers in ,1 Then a fuzzy subset ''s'' : ''S'' $\rightarrow$ ,1of 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 ''s''(''x'') = lim ''h''(''x'',''n''). We say that ''s'' is ''decidable'' if both ''s'' and its complement –''s'' are recursively enumerable. An extension of such a theory to the general case of the L-subsets is possible (see Gerla 2006). The proposed definitions are well related to fuzzy logic. Indeed, the following theorem holds true (provided that the deduction apparatus of the considered fuzzy logic satisfies some obvious effectiveness property). Any "axiomatizable" fuzzy theory is recursively enumerable. In particular, the
fuzzy set In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, defined ...
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 support for a "Church thesis" for
fuzzy mathematics Fuzzy mathematics is the branch of mathematics including fuzzy set theory and fuzzy logic that deals with partial inclusion of elements in a set on a spectrum, as opposed to simple binary "yes" or "no" (0 or 1) inclusion. It started in 1965 ...
, the proposed notion of recursive enumerability for fuzzy subsets is the adequate one. In order to solve this, an extension of the notions of fuzzy grammar and fuzzy
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...
are necessary. Another open question is to start from this notion to find an extension of Gödel's theorems to fuzzy logic.

# Compared to other logics

## Probability

Fuzzy logic and probability address different forms of uncertainty. While both fuzzy logic and probability theory can represent degrees of certain kinds of subjective belief,
fuzzy set theory In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, defined ...
uses the concept of fuzzy set membership, i.e., how much an observation is within a vaguely defined set, and probability theory uses the concept of subjective probability, i.e., frequency of occurrence or likelihood of some event or condition . The concept of fuzzy sets was developed in the mid-twentieth century at Berkeley as a response to the lack of a probability theory for jointly modelling uncertainty and
vagueness In linguistics and philosophy, a vague predicate is one which gives rise to borderline cases. For example, the English adjective "tall" is vague since it is not clearly true or false for someone of middling height. By contrast, the word "prime" is ...
.
Bart Kosko Bart Andrew Kosko (born February 7, 1960) is a writer and professor of electrical engineering and law at the University of Southern California (USC). He is a researcher and popularizer of fuzzy logic, neural networks, and noise, and author of sev ...
claims in Fuzziness vs. Probability that probability theory is a subtheory of fuzzy logic, as questions of degrees of belief in mutually-exclusive set membership in probability theory can be represented as certain cases of non-mutually-exclusive graded membership in fuzzy theory. In that context, he also derives Bayes' theorem from the concept of fuzzy subsethood. Lotfi A. Zadeh argues that fuzzy logic is different in character from probability, and is not a replacement for it. He fuzzified probability to fuzzy probability and also generalized it to
possibility theory Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. It uses measures of possibility and necessity between 0 and 1, ranging from impossible to possible and unnecessa ...
. More generally, fuzzy logic is one of many different extensions to classical logic intended to deal with issues of uncertainty outside of the scope of classical logic, the inapplicability of probability theory in many domains, and the paradoxes of
Dempster–Shafer theory The theory of belief functions, also referred to as evidence theory or Dempster–Shafer theory (DST), is a general framework for reasoning with uncertainty, with understood connections to other frameworks such as probability, possibility and i ...
.

## Ecorithms

Computational theorist
Leslie Valiant Leslie Gabriel Valiant (born 28 March 1949) is a British American computer scientist and computational theorist. He was born to a chemical engineer father and a translator mother. He is currently the T. Jefferson Coolidge Professor of Compu ...
uses the term ''ecorithms'' to describe how many less exact systems and techniques like fuzzy logic (and "less robust" logic) can be applied to learning algorithms. Valiant essentially redefines machine learning as evolutionary. In general use, ecorithms are algorithms that learn from their more complex environments (hence ''eco-'') to generalize, approximate and simplify solution logic. Like fuzzy logic, they are methods used to overcome continuous variables or systems too complex to completely enumerate or understand discretely or exactly. Ecorithms and fuzzy logic also have the common property of dealing with possibilities more than probabilities, although feedback and feed forward, basically stochastic weights, are a feature of both when dealing with, for example,
dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
s.

## Gödel G∞ logic

Another logical system where truth values are real numbers between 0 and 1 and where AND & OR operators are replaced with MIN and MAX is Gödel's G logic. This logic has many similarities with fuzzy logic but defines negation differently and has an internal implication. Negation $\neg_G$ and implication
intuitionistic logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
, making it particularly well-behaved among all possible choices of logical systems with real numbers between 0 and 1 as truth values. In this case, implication may be interpreted as "x is less true than y" and negation as "x is less true than 0" or "x is strictly false", and for any $x$ and $y$, we have that $AND\left(x, x \mathrel y\right) = AND\left(x,y\right)$. In particular, in Gödel logic negation is no longer an involution and double negation maps any nonzero value to 1.

# Compensatory fuzzy logic

Compensatory fuzzy logic (CFL) is a branch of fuzzy logic with modified rules for conjunction and disjunction. When the truth value of one component of a conjunction or disjunction is increased or decreased, the other component is decreased or increased to compensate. This increase or decrease in truth value may be offset by the increase or decrease in another component. An offset may be blocked when certain thresholds are met. Proponents claim that CFL allows for better computational semantic behaviors and mimic natural language. According to Jesús Cejas Montero (2011) The Compensatory fuzzy logic consists of four continuous operators: conjunction (c); disjunction (d); fuzzy strict order (or); and negation (n). The conjunction is the geometric mean and its dual as conjunctive and disjunctive operators.

# Markup language standardization

The IEEE 1855, the IEEE STANDARD 1855–2016, is about a specification language named
Fuzzy Markup Language Fuzzy Markup Language (FML) is a specific purpose markup language based on XML, used for describing the structure and behavior of a fuzzy system independently of the hardware architecture devoted to host and run it. Overview FML was designed and ...
(FML) developed by the IEEE Standards Association. FML allows modelling a fuzzy logic system in a human-readable and hardware independent way. FML is based on eXtensible Markup Language (
XML Extensible Markup Language (XML) is a markup language and file format for storing, transmitting, and reconstructing arbitrary data. It defines a set of rules for encoding documents in a format that is both human-readable and machine-readable ...
). The designers of fuzzy systems with FML have a unified and high-level methodology for describing interoperable fuzzy systems. IEEE STANDARD 1855–2016 uses the
W3C The World Wide Web Consortium (W3C) is the main international standards organization for the World Wide Web. Founded in 1994 and led by Tim Berners-Lee, the consortium is made up of member organizations that maintain full-time staff working to ...
XML Schema An XML schema is a description of a type of XML document, typically expressed in terms of constraints on the structure and content of documents of that type, above and beyond the basic syntactical constraints imposed by XML itself. These constra ...
definition language to define the syntax and semantics of the FML programs. Prior to the introduction of FML, fuzzy logic practitioners could exchange information about their fuzzy algorithms by adding to their software functions the ability to read, correctly parse, and store the result of their work in a form compatible with the Fuzzy Control Language (FCL) described and specified by Part 7 of IEC 61131.

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Adaptive neuro fuzzy inference system An adaptive neuro-fuzzy inference system or adaptive network-based fuzzy inference system (ANFIS) is a kind of artificial neural network that is based on Takagi–Sugeno fuzzy inference system. The technique was developed in the early 1990s. Since ...
(ANFIS) *
Artificial neural network Artificial neural networks (ANNs), usually simply called neural networks (NNs) or neural nets, are computing systems inspired by the biological neural networks that constitute animal brains. An ANN is based on a collection of connected unit ...
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Defuzzification Defuzzification is the process of producing a quantifiable result in crisp logic, given fuzzy sets and corresponding membership degrees. It is the process that maps a fuzzy set to a crisp set. It is typically needed in fuzzy control systems. Th ...
* Expert system * False dilemma *
Fuzzy architectural spatial analysis Fuzzy architectural spatial analysis (FASA) (also fuzzy inference system (FIS) based architectural space analysis or fuzzy spatial analysis) is a spatial analysis method of analysing the spatial formation and architectural space intensity within ...
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Fuzzy classification Fuzzy classification is the process of grouping elements into fuzzy sets whose membership functions are defined by the truth value of a fuzzy propositional function. A fuzzy propositional function is analogous toRussel, B. (1919). ''Introduction to ...
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Fuzzy concept A fuzzy concept is a kind of concept of which the boundaries of application can vary considerably according to context or conditions, instead of being fixed once and for all. This means the concept is vague in some way, lacking a fixed, precise me ...
* Fuzzy Control Language * Fuzzy control system * Fuzzy electronics *
Fuzzy subalgebra Fuzzy subalgebras theory is a chapter of fuzzy set theory. It is obtained from an interpretation in a multi-valued logic of axioms usually expressing the notion of subalgebra of a given algebraic structure. Definition Consider a first order l ...
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FuzzyCLIPS FuzzyCLIPS is a fuzzy logic extension of the CLIPS (C Language Integrated Production System) expert system shell from NASA. It was developed by the Integrated Reasoning Group of the Institute for Information Technology of the National Research Cou ...
* High Performance Fuzzy Computing * IEEE Transactions on Fuzzy Systems *
Interval finite element In numerical analysis, the interval finite element method (interval FEM) is a finite element method that uses interval parameters. Interval FEM can be applied in situations where it is not possible to get reliable probabilistic characteristics of ...
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Machine learning Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine ...
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Neuro-fuzzy In the field of artificial intelligence, neuro-fuzzy refers to combinations of artificial neural networks and fuzzy logic. Overview Neuro-fuzzy hybridization results in a hybrid intelligent system that these two techniques by combining the human ...
* Noise-based logic * Rough set *
Sorites paradox The sorites paradox (; sometimes known as the paradox of the heap) is a paradox that results from vague predicates. A typical formulation involves a heap of sand, from which grains are removed individually. With the assumption that removing a sing ...
* Type-2 fuzzy sets and systems *
Vector logic Vector logicMizraji, E. (1992)Vector logics: the matrix-vector representation of logical calculus.Fuzzy Sets and Systems, 50, 179–185Mizraji, E. (2008Vector logic: a natural algebraic representation of the fundamental logical gates.Journal of Logi ...

# Bibliography

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Formal fuzzy logic
– article at
Citizendium Citizendium ( ; "the citizens' compendium of everything") is an English-language wiki-based free online encyclopedia launched by Larry Sanger, co-founder of Nupedia and Wikipedia. It was first announced in September 2006 as a fork of the Engli ...

IEC 1131-7 CD1
IEC 1131-7 CD1 PDF
Fuzzy Logic
– article at
Scholarpedia ''Scholarpedia'' is an English-language wiki-based online encyclopedia with features commonly associated with open-access online academic journals, which aims to have quality content in science and medicine. ''Scholarpedia'' articles are written ...

Modeling With Words
– article at Scholarpedia
Fuzzy logic
– article at Stanford Encyclopedia of Philosophy
Fuzzy Math
– Beginner level introduction to Fuzzy Logic

– Fuzziness in everyday life, science, religion, ethics, politics, etc.
Fuzzylite
– A cross-platform, free open-source Fuzzy Logic Control Library written in C++. Also has a very useful graphic user interface in QT4.
More Flexible Machine Learning
– MIT describes one application.
Semantic Similarity
MIT provides details about fuzzy semantic similarity. {{DEFAULTSORT:Fuzzy Logic Logic in computer science Non-classical logic Probability interpretations Iranian inventions Azerbaijani inventions