# Fuzzy logic problems and solutions pdf

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- Fuzzy logic
- Fuzzy Logic Tutorial: What is, Architecture, Application, Example
- Fuzzy Implications: Some Recently Solved Problems

*Fuzzy logic is a multi-valued logic which is similar to human thinking and interpretation. It has the potential of combining human heuristics into computer-assisted decision making, which is applicable to individual patients as it takes into account all the factors and complexities of individuals. Fuzzy logic has been applied in all disciplines of medicine in some form and recently its applicability in neurosciences has also gained momentum.*

Many researchers study fuzzy logic [ 1 , 2 ]. Zadeh [ 3 ] and Dubois and Prade [ 4 ] introduced fuzzy number and fuzzy arithmetic. Firstly, Chang and Zadeh introduced the concept of fuzzy derivative [ 6 ]. Dubois and Prade [ 7 ] followed up their approach. Other methods were studied in several papers [ 8 , 9 , 10 , 11 , 12 ].

## Fuzzy logic

In fuzzy mathematics , fuzzy logic is a form of many-valued logic in which the true values of variables may be any real number between 0 and 1 both inclusive. It is employed to handle the concept of partial truth, where the true value may range between completely true and completely false. The term fuzzy logic was introduced with the proposal of fuzzy set theory by Lotfi Zadeh.

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 and imprecise information hence the term fuzzy. These models have the capability of recognising, representing, manipulating, interpreting, and utilising data and information that are vague and lack certainty. Fuzzy logic has been applied to many fields, from control theory to artificial intelligence.

Classical logic only permits conclusions which 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 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 is a mathematical model of ignorance.

A basic application might characterize various sub-ranges 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.

While variables in mathematics usually take numerical values, 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 or adverbs.

For example, we can use the hedges rather and somewhat to construct the additional values rather old or somewhat young. Fuzzification operations can map mathematical input values into fuzzy membership functions. And the opposite de-fuzzifying operations can be used to map a fuzzy output membership function into a "crisp" output value that can be then used for decision or control purposes. 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 [0,1]. 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. 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, [11] 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. The generalization of AND is known as a t-norm. 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. 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 [ citation needed ].

It has also been used in recognition of hand-written symbols in Sony pocket computers, flight aid for helicopters, controlling of subway systems in order to improve driving comfort, precision of halting, and power economy, improved fuel consumption for automobiles, single-button control for washing machines, automatic motor control for vacuum cleaners with recognition of surface condition and degree of soiling, and prediction systems for early recognition of earthquakes through the Institute of Seismology Bureau of Meteorology, Japan.

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.

This is even more challenging when one has to elicit such data from humans usually, patients. As it said "The envelope of what can be achieved and what cannot be achieved in medical diagnosis, ironically, is itself a fuzzy one" [Seven Challenges, ].

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. 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 quantifier in t-norm fuzzy logics is the infimum of the truth degrees of the instances of the quantified subformula while the semantics of the existential quantifier is the supremum of the same.

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. 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. Fuzzy querying languages have been defined, such as the SQLf by P. Bosc et al. 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.

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 uses the concept of fuzzy set membership, i. The concept of fuzzy sets was developed in the mid-twentieth century at Berkeley [21] as a response to the lacking of probability theory for jointly modelling uncertainty and vagueness.

Bart Kosko claims in Fuzziness vs. Probability [23] 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.

## Fuzzy Logic Tutorial: What is, Architecture, Application, Example

Fuzzy Logic is defined as a many-valued logic form which may have truth values of variables in any real number between 0 and 1. It is the handle concept of partial truth. In real life, we may come across a situation where we can't decide whether the statement is true or false. At that time, fuzzy logic offers very valuable flexibility for reasoning. Fuzzy logic algorithm helps to solve a problem after considering all available data. Then it takes the best possible decision for the given the input. The FL method imitates the way of decision making in a human which consider all the possibilities between digital values T and F.

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have their problems (some numbers are not available and hard to define, the real So, they do not represent the definite solution to deal with uncertainty in the real Table Applications of fuzzy logic in Japan and Korea (fielded products).

## Fuzzy Implications: Some Recently Solved Problems

Harpreet Singh, Madan M. Solo, Lotfi A. Box , Beijing , China. This special issue is dedicated to Professor Lotfi A. Zadeh, the father of fuzzy logic.

Беккер все еще не мог прийти в себя от всего, что услышал. - Может, там был кто-нибудь. - Нет.

Проваливай и умри. Он повернулся к Росио и заговорил с ней по-испански: - Похоже, я злоупотребил вашим гостеприимством. - Не обращайте на него внимания, - засмеялась. - Он просто расстроен. Но он получит то, что ему причитается.

*Через некоторое время им с огромным трудом удалось расширить щель до одного фута. - Не отпускай, - сказал Стратмор, стараясь изо всех сил.*

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- Фонтейн, как обычно, говорил спокойно и деловито. Глаза Джаббы по-прежнему выражали шок и растерянность, когда сзади раздался душераздирающий крик: - Джабба. Джабба. Это кричала Соши Кута, его технический ассистент, подбегая к платформе с длиннющей распечаткой в руке. У нее был такой вид, словно она только что увидела призрак. - Джабба! - Соши задыхалась. - Червь… я знаю, на что он запрограммирован! - Она сунула распечатку Джаббе.