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Since the red arrow points to zero, this temperature may be interpreted as "not hot"; i. The orange arrow pointing at 0. Therefore, this temperature has 0. The degree of membership assigned for each fuzzy set is the result of fuzzification. Fuzzy sets 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. Fuzzy logic works with membership values in a way that mimics Boolean logic.

There are several ways to this. A common replacement is called the Zadeh operators :. 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. However, an arbitrary choice table does not always define a fuzzy logic function.

In the paper, [10] 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.

Given a certain temperature, the fuzzy variable hot has a certain truth value, which is copied to the high variable. The goal is to get a continuous variable from fuzzy truth values.

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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. 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. Many of the early successful applications of fuzzy logic were implemented in Japan. The first notable application was on the subway train in Sendai , in which fuzzy logic was able to improve the economy, comfort, and precision of the ride. Fuzzy logic is an important concept when it comes to 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. One of the common application areas that use fuzzy logic is computer-aided diagnosis CAD in medicine. Fuzzy logic can be highly appropriate to describe key characteristics of this lesion.

Fuzzy logic can be used in many different aspects within the CAD framework. 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.

## Decision-Making in a Fuzzy Environment

This is even more challenging when one has to elicit such data from humans usually, patients. 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 CAD application area but still requires more research to achieve its full potential.

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In mathematical logic , there are several formal systems of "fuzzy logic", most of which are in the family of t-norm 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.

The notions of a "decidable subset" and " recursively enumerable subset" are basic ones for classical mathematics and classical logic. Thus the question of a suitable extension of them to fuzzy set theory is a crucial one. A first proposal in such a direction was made by E. Santos by the notions of fuzzy Turing machine , Markov normal fuzzy algorithm and fuzzy program see Santos Successively, L.

Biacino and G. Gerla argued that the proposed definitions are rather questionable.

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For example, in [20] 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.

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 The proposed definitions are well related with 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 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 mathematics , 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 are necessary. Once fuzzy relations are defined, it is possible to develop fuzzy relational databases. Medina, M. Vila et al.