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.[11]
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.
However, an arbitrary choice table does not always define a fuzzy logic function. In the paper (Zaitsev, et al),[16] 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 TSK system[17] 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:
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 systems.
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. A first notable application was on the Sendai Subway 1000 series, in which fuzzy logic was able to improve the economy, comfort, and precision of the ride. It has also been used for handwriting recognition 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.[18]
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.
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 .mw-parser-output .templatequoteoverflow:hidden;margin:1em 0;padding:0 40px.mw-parser-output .templatequote .templatequoteciteline-height:1.5em;text-align:left;padding-left:1.6em;margin-top:0
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.[25] 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.
One of the common application areas that use fuzzy logic is image-based computer-aided diagnosis (CAD) in medicine.[26] 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.
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.
Similar to the way predicate logic is created from propositional logic, predicate fuzzy logics extend fuzzy systems by universal and existential quantifiers. 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.
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 support 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. Another open question is to start from this notion to find an extension of GÃdel's theorems to fuzzy logic.
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.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[clarification needed]. The concept of fuzzy sets was developed in the mid-twentieth century at Berkeley [28] as a response to the lack of a probability theory for jointly modelling uncertainty and vagueness.[29]
Bart Kosko claims in Fuzziness vs. Probability[30] 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.[31]
Computational theorist Leslie Valiant 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.[32] 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 systems.
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[who?] claim that CFL allows for better computational semantic behaviors and mimic natural language.[vague][33][34]
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