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In logicfuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number 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, dissertation thesis and research project workbook.
The term fuzzy logic was introduced with the proposal of fuzzy set theory by Azerbaijani scientist 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 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 dissertation thesis and research project workbook instances, the truth appears as the result of reasoning from inexact or partial knowledge in which the sampled answers are mapped on a spectrum. Dissertation thesis and research project workbook 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 vaguenesswhile 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. 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 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 dissertation thesis and research project workbook 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 coldwarmand hot are represented by functions mapping a temperature scale, dissertation thesis and research project workbook.
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. this temperature has zero membership in the fuzzy set "hot", dissertation thesis and research project workbook.
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 dissertation thesis and research project workbook, 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. 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 Zadeh operators :. There are also other operators, more linguistic in nature, called hedges that can be applied. These are generally adverbs such as veryor somewhatwhich 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, [13] 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. Should an output variable occur in several THEN parts, then the values from the respective IF parts are combined using the OR operator. 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 dissertation thesis and research project workbook 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 Sendaiin 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, dissertation thesis and research project workbook. 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 logicthere 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 dissertation thesis and research project workbook 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 machineMarkov normal fuzzy algorithm and fuzzy program see Santos Dissertation thesis and research project workbook, L. Biacino and G.
Gerla argued that the proposed definitions are rather questionable. For example, in [22] 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 [0,1]. We say that s is decidable if both s and its complement — s are recursively enumerable. Dissertation thesis and research project workbook 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, dissertation thesis and research project workbook, 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 support for a "Church thesis" for fuzzy mathematicsthe 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. Another open question is to start from this notion to find an extension of Gödel 's theorems to fuzzy logic. 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 Buckles-Petry model, the Prade-Testemale Model, the Umano-Fukami model or the GEFRED model by J.
Medina, M. 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.
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