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Fuzzy logic is a form of multivalued logic derived from fuzzy set theory to deal with reasoning that is approximate rather than precise. In contrast with "crisp logic", where binary sets have binary logic, the fuzzy logic variables may have a membership value of not only 0 or 1 – that is, the degree of truth of a statement can range between 0 and 1 and is not constrained to the two truth values of classic propositional logic.^{[1]} Furthermore, when linguistic variables are used, these degrees may be managed by specific functions.
Fuzzy logic emerged as a consequence of the 1965 proposal of fuzzy set theory by Lotfi Zadeh.^{[2]}^{[3]} Though fuzzy logic has been applied to many fields, from control theory to artificial intelligence, it still remains controversial among most statisticians, who prefer Bayesian logic, and some control engineers, who prefer traditional twovalued logic.
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Fuzzy logic and probabilistic logic are mathematically similar – both have truth values ranging between 0 and 1 – but conceptually distinct, due to different interpretations. Fuzzy logic corresponds to "degrees of truth", while probabilistic logic corresponds to "probability, likelihood"; as these differ, fuzzy logic and probabilistic logic yield different models of the same realworld situations.
Both degrees of truth and probabilities range between 0 and 1 and hence may seem similar at first. Truth represents membership in vaguely defined sets, not likelihood of some event or condition as in probability theory. For example, let a 100 ml glass contain 30 ml of water. Then we may consider two concepts: Empty and Full. The meaning of each of them can be represented by a certain fuzzy set. Then 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. It is essential to realize that fuzzy logic uses truth degrees as a mathematical model of the vagueness phenomenon while probability is a mathematical model of randomness. 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. This model, however, has no sense without accepting occurrence of some event, e.g. that after a few minutes, the glass will be half empty. 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. Consequently, probability has nothing in common with fuzziness, these are simply different concepts which superficially seem similar because of using the same unit interval of real numbers [0,1]. Still, since theorems such as De Morgan's have dual applicability and properties of random variables are analogous to properties of binary logic states, one can see where the confusion might arise.
A basic application might characterize subranges of a continuous variable. For instance, a temperature measurement for antilock 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, the meaning of the expressions cold, warm, and hot is 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". The orange arrow (pointing at 0.2) may describe it as "slightly warm" and the blue arrow (pointing at 0.8) "fairly cold".
While variables in mathematics usually take numerical values, in fuzzy logic applications, the nonnumeric linguistic variables are often used to facilitate the expression of rules and facts.^{[4]}
A linguistic variable such as age may have a value such as young or its antonym old. However, the great utility of linguistic variables is that they can be modified via linguistic hedges applied to primary terms. The linguistic hedges can be associated with certain functions. For example, L. A. Zadeh proposed to take the square of the membership function. This model, however, does not work properly. For more details, see the references.
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 IFTHEN rules, or constructs that are equivalent, such as fuzzy associative matrices.
Rules are usually expressed in the form:
IF variable IS property THEN action
For example, a 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
There is no "ELSE" – all of the rules are evaluated, because the temperature might be "cold" and "normal" at the same time to different 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. So for the fuzzy variables x and y:
NOT x = (1  truth(x))
x AND y = minimum(truth(x), truth(y))
x OR y = maximum(truth(x), truth(y))
There are also other operators, more linguistic in nature, called hedges that can be applied. These are generally adverbs such as "very", or "somewhat", which modify the meaning of a set using a mathematical formula.
In application, the programming language Prolog is well geared to implementing fuzzy logic^{[citation needed]} with its facilities to set up a database of "rules" which are queried to deduct logic. This sort of programming is known as logic programming.
Once fuzzy relations are defined, it is possible to develop fuzzy relational databases. The first fuzzy relational database, FRDB, appeared in Maria Zemankova's dissertation. Later, some other models arose like the BucklesPetry model, the PradeTestemale Model, the UmanoFukami 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.
In mathematical logic, there are several formal systems of "fuzzy logic"; most of them belong among socalled tnorm fuzzy logics.
The most important propositional fuzzy logics are:
These extend the abovementioned 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 tnorm fuzzy logics is the infimum (resp. supremum) of the truth degrees of the instances of the quantified subformula.
These logics, called fuzzy type theories, extend predicate fuzzy logics to be able to quantify also predicates and higher order objects. A fuzzy type theory is a generalization of classical simple type theory introduced by B. Russell ^{[5]} and mathematically elaborated by A. Church ^{[6]} and L. Henkin^{[7]}.
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 (see Santos 1970). Successively, L. Biacino and G. Gerla showed that such a definition is not adequate and therefore proposed the following one. Ü denotes the set of rational numbers in [0,1]. A fuzzy subset s : S [0,1] of a set S is recursively enumerable if a recursive map h : S×N 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 Lsubsets is proposed in Gerla 2006. The proposed definitions are well related with fuzzy logic. Indeed, the following theorem holds true (provided that the deduction apparatus of the fuzzy logic satisfies some obvious effectiveness property).
Theorem. Any axiomatizable fuzzy theory is recursively enumerable. In particular, the fuzzy set of logically true formulas is recursively enumerable in spite of the fact that the crisp set of valid formulas is not recursively enumerable, in general. Moreover, any axiomatizable and complete theory is decidable.
It is an open question to give supports for a Church thesis for fuzzy logic claiming that the proposed notion of recursive enumerability for fuzzy subsets is the adequate one. To this aim, further investigations on the notions of fuzzy grammar and fuzzy Turing machine should be necessary (see for example Wiedermann's paper). Another open question is to start from this notion to find an extension of Gödel’s theorems to fuzzy logic.
Fuzzy logic is used in the operation or programming of:
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Fuzzy logic is a sort of computer logic that is different from boolean algebra. It is different in the way that it allows values to be more accurate than on or off. While boolean logic only allows true or false, fuzzy logic allows all things in between. An example of this could be a computer game: A person is standing in a doorway while a thing explodes. The character is hit or not hit if boolean logic is used, but the doorway protects him from the explosion. Therefore, he might only be hit 20%, and takes less damage.
To put it in more precise mathematical terms, classical logic has two values. These values are usually called false (0) or true (1). With fuzzy logic, a (calculated) value of 0.8 or 0.971 is possible. It is important to know the difference between fuzzy logic and chance. A coin that is thrown has a chance of 0.5 for landing heads up. If it is thrown 1000 times, it is expected that it will land with heads side up 500 times. With fuzzy logic, a thing with a "truth value" of 0.5 will have a value of 0.5 no matter how many times it is done. It is not a 50% chance of true or untrue, it is 50% true and 50% untrue at the same time. Fuzzy logic is used a lot in expert systems and neuronal networks.
Humans tend to use a combination of predicate logic and fuzzy logic. If you are an outfielder catching a baseball hit into the air; then your precise logic will calculate trajectory and start you running to the point of intercept (catching). However, once close to the ball the eyes & brain of the outfielder lacks the ability to accurately estimate distance and speed because the ball is coming straight at the outfielder. The human brain switches to fuzzy logic that says "get me closer", "get me closer", and so on. That is why you see outfielders in baseball run to a spot and then move around as the ball gets closer.
Predicate logic says calculate the point to be at to catch the ball. Fuzzy logic says because of wind or other things you might not be in the correct place so just keep getting closer until you catch the ball.
In predicate logic it is the mathematics of calculating the path of the ball that determines your action. In fuzzy logic it is the error of your calculations that determines your action.
In effect, it's like your brain trying to control your finger to pick your nose while traveling down a bumpy road.
