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fuzzy logic

Fuzzy logic Fuzzy logic has been extended to handle the concept of partial truth, where the truth value may range between completely true and completely false

Fuzzy logic The term "fuzzy logic" was introduced with the 1965 proposal of fuzzy set theory by Lotfi A. Zadeh. Fuzzy logic has been applied to many fields, from control theory to Artificial Intelligence. Fuzzy logics however had been studied since the 1920s as infinite-valued logics notably by Łukasiewicz and Tarski.

Fuzzy logic - Overview 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 ignorance.

Fuzzy logic - Linguistic variables
While variables in mathematics usually take numerical values, in fuzzy logic applications, the non-numeric linguistic variables are often used to facilitate the expression of rules and facts.

Fuzzy logic - Early applications
The Japanese were the first to utilize fuzzy logic for practical applications. The first notable application was on the high-speed train in Sendai, in which fuzzy logic was able to improve the economy, comfort, and precision of the ride. It has also been used in recognition of hand written symbols in Sony pocket computers, Canon auto-focus technology, Omron auto-aiming cameras, earthquake prediction and modeling at the Institute of Seismology Bureau of Metrology in Japan, etc.

Fuzzy logic - Hard science with IF-THEN rules
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 IF-THEN rules, or constructs that are equivalent, such as fuzzy associative matrices.

Fuzzy logic - Hard science with IF-THEN rules
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:

Fuzzy logic - Logical analysis
In mathematical logic, there are several formal systems of "fuzzy logic"; most of them belong among so-called t-norm fuzzy logics.

Fuzzy logic - Propositional fuzzy logics
The most important propositional fuzzy logics are:

Monoidal t-norm-based propositional fuzzy logic MTL is an axiomatization of logic where conjunction is defined by a left continuous t-norm, and implication is defined as the residuum of the t-norm. Its models correspond to MTL-algebras that are prelinear commutative bounded integral residuated lattices.

Basic propositional fuzzy logic BL is an extension of MTL logic where conjunction is defined by a continuous t-norm, and implication is also defined as the residuum of the t-norm. Its models correspond to BL-algebras.

Łukasiewicz fuzzy logic is the extension of basic fuzzy logic BL where standard conjunction is the Łukasiewicz t-norm. It has the axioms of basic fuzzy logic plus an axiom of double negation, and its models correspond to MV-algebras.

Gödel fuzzy logic is the extension of basic fuzzy logic BL where conjunction is Gödel t-norm. It has the axioms of BL plus an axiom of idempotence of conjunction, and its models are called G-algebras.

Product fuzzy logic is the extension of basic fuzzy logic BL where conjunction is product t-norm. It has the axioms of BL plus another axiom for cancellativity of conjunction, and its models are called product algebras.

Fuzzy logic with evaluated syntax (sometimes also called Pavelka's logic), denoted by EVŁ, is a further generalization of mathematical fuzzy logic. While the above kinds of fuzzy logic have traditional syntax and many-valued semantics, in EVŁ is evaluated also syntax. This means that each formula has an evaluation. Axiomatization of EVŁ stems from Łukasziewicz fuzzy logic. A generalization of classical Gödel completeness theorem is provable in EVŁ.

Fuzzy logic - Predicate 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. existential) quantifier in t-norm fuzzy logics is the infimum (resp. supremum) of the truth degrees of the instances of the quantified subformula.

Fuzzy logic - Decidability issues for fuzzy logic
Indeed, the following theorem holds true (provided that the deduction apparatus of the considered fuzzy logic satisfies some obvious effectiveness property).

Fuzzy logic - Decidability issues for fuzzy logic
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. To this aim, an extension of 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 - Synthesis of fuzzy logic functions given in tabular form
The task of synthesis of fuzzy logic function given in tabular form was solved in. New concepts of constituents of minimum and maximum were introduced. The sufficient and necessary conditions that a choice table defines a fuzzy logic function were derived.

Fuzzy logic - Comparison to probability
(cf.) More generally, fuzzy logic is one of many different proposed extensions to classical logic, known as probabilistic logics, intended to deal with issues of uncertainty in classical logic, the inapplicability of probability theory in many domains, and the paradoxes of Dempster-Shafer theory.

Fuzzy logic - Relation to ecorithms
Ecorithms and fuzzy logic also have the common property of dealing with possibilities more than probabilities, although feedback and feedforward, basically stochastic "weights," are a feature of both when dealing with, for example, dynamical systems.

Fuzzy logic - Relation to ecorithms
58 of the reference comparing induction/invariance, robust, mathematical and other logical limits in computing, where techniques including fuzzy logic and natural data selection (ala "computational Darwinism") can be used to shortcut computational complexity and limits in a "practical" way (such as the brake temperature example in this article).

Fuzzy logic - Bibliography
Biacino, L.; Gerla, G. (2002). "Fuzzy logic, continuity and effectiveness". Archive for Mathematical Logic 41 (7): 643–667. doi: /s ISSN

Hájek, Petr (1998). Metamathematics of fuzzy logic. Dordrecht: Kluwer. ISBN

Hájek, Petr (1995). "Fuzzy logic and arithmetical hierarchy". Fuzzy Sets and Systems 3 (8): 359–363. doi: / (94)00299-M. ISSN

Klir, George J.; Yuan, Bo (1995). Fuzzy sets and fuzzy logic: theory and applications. Upper Saddle River, NJ: Prentice Hall PTR. ISBN

Kosko, Bart (1993). Fuzzy thinking: the new science of fuzzy logic. New York: Hyperion. ISBN X.

Kosko, Bart; Isaka, Satoru (July 1993). "Fuzzy Logic". Scientific American 269 (1): 76–81. doi: /scientificamerican

Lohani, A.K.; Goel, N.K.; Bhatia K.K.S. (2007). "Deriving stage–discharge–sediment concentration relationships using fuzzy logic". Hydrological Sciences Journal 52 (4): 793–807. doi: /hysj

Lohani, A.K.; Goel, N.K.; Bhatia K.K.S. (2011). "Comparative study of neural network, fuzzy logic and linear transfer function techniques in daily rainfall‐runoff modelling under different input domains". Hydrological Processes 25 (2): 175–193. doi: /hyp.7831.

Montagna, F. (2001). "Three complexity problems in quantified fuzzy logic". Studia Logica 68 (1): 143–152. doi: /A: ISSN

Novák, Vilém; Perfilieva, Irina; Močkoř, Jiří (1999). Mathematical principles of fuzzy logic. Dordrecht: Kluwer Academic. ISBN

Steeb, Willi-Hans (2008). The Nonlinear Workbook: Chaos, Fractals, Cellular Automata, Neural Networks, Genetic Algorithms, Gene Expression Programming, Support Vector Machine, Wavelets, Hidden Markov Models, Fuzzy Logic with C++, Java and SymbolicC++ Programs: 4edition. World Scientific. ISBN

Tsitolovsky, Lev; Sandler, Uziel (2008). Neural Cell Behavior and Fuzzy Logic. Springer. ISBN ISBN

Van Pelt, Miles (2008). Fuzzy Logic Applied to Daily Life. Seattle, WA: No No No No Press. ISBN

Von Altrock, Constantin (1995). Fuzzy logic and NeuroFuzzy applications explained. Upper Saddle River, NJ: Prentice Hall PTR. ISBN

Control system - Fuzzy logic
Fuzzy logic is an attempt to apply the easy design of logic controllers to the control of complex continuously-varying systems. Basically, a measurement in a fuzzy logic system can be partly true, that is if yes is 1 and no is 0, a fuzzy measurement can be between 0 and 1.

The rules of the system are written in natural language and translated into fuzzy logic. For example, the design for a furnace would start with: If the temperature is too high, reduce the fuel to the furnace. If the temperature is too low, increase the fuel to the furnace.

Fuzzy logic, then, modifies Boolean logic to be arithmetical. Usually the not operation is output = 1 - input, the and operation is output = input.1 multiplied by input.2, and or is output = 1 - ((1 - input.1) multiplied by (1 - input.2)). This reduces to Boolean arithmetic if values are restricted to 0 and 1, instead of allowed to range in the unit interval .

When a robust fuzzy design is reduced into a single, quick calculation, it begins to resemble a conventional feedback loop solution and it might appear that the fuzzy design was unnecessary. However, the fuzzy logic paradigm may provide scalability for large control systems where conventional methods become unwieldy or costly to derive.

Fuzzy electronics is an electronic technology that uses fuzzy logic instead of the two-value logic more commonly used in digital electronics.

Anomaly detection - Fuzzy Logic based method
In the last years Fuzzy Logic has been adopted in several outlier detection approaches in order to improve results coming from popular outlier detection techniques.

Anomaly detection - Fuzzy Logic based method
Yousri et al. proposed an approach based on fuzzy logic in order to merge results obtained with an outlier detection method, which estabilishes if a pattern is an outlier and a clustering algorithm which provides results in allocating patterns to clusters. The two results, provided by the two different approaches, are then combined to give a meaure of outlierness.

Vagueness - Fuzzy logic
Advocates of the fuzzy logic approach have included K

Lukasiewicz fuzzy logic
Sci. et Lettres Varsovie Cl. III 23, 30–50 (1930). It belongs to the classes of t-norm fuzzy logicsHájek P., 1998, Metamathematics of Fuzzy Logic. Dordrecht: Kluwer. and substructural logics.Ono, H., 2003, Substructural logics and residuated lattices — an introduction. In F.V. Hendricks, J. Malinowski (eds.): Trends in Logic: 50 Years of Studia Logica, Trends in Logic '20': 177–212.

Lukasiewicz fuzzy logic - General algebraic semantics
Like other t-norm fuzzy logics, propositional infinite-valued Łukasiewicz logic enjoys completeness with respect to the class of all algebras for which the logic is sound (that is, MV-algebras) as well as with respect to only linear ones. This is expressed by the general, linear, and standard completeness theorems:

T-norm fuzzy logics 'T-norm fuzzy logics' are a family of non-classical logics, informally delimited by having a semantics which takes the real unit interval [0,1] for the system of truth values and functions called t-norms for permissible interpretations of logical conjunction|conjunction. They are mainly used in applied fuzzy logic and fuzzy set|fuzzy set theory as a theoretical basis for approximate reasoning.

T-norm fuzzy logics Both propositional logic|propositional and first-order logic|first-order (or higher-order logic|higher-order) t-norm fuzzy logics, as well as their expansions by Modal operator|modal and other operators, are studied

T-norm fuzzy logics Some independently motivated logics belong among t-norm fuzzy logics, too, for example Łukasiewicz logic (which is the logic of the Łukasiewicz t-norm) or intermediate logic|Gödel–Dummett logic (which is the logic of the minimum t-norm).

T-norm fuzzy logics - Motivation
In propositional t-norm fuzzy logics, propositional formula|propositional connectives are stipulated to be truth-functional, that is, the truth value of a complex proposition formed by a propositional connective from some constituent propositions is a function (called the truth function of the connective) of the truth values of the constituent propositions

T-norm fuzzy logics impose certain natural constraints on the truth function of logical conjunction|conjunction. The truth function *\colon[0,1]^2\to[0,1] of conjunction is assumed to satisfy the following conditions:

This condition, among other things, ensures a good behavior of (residual) implication derived from conjunction; to ensure the good behavior, however, left-continuity (in either argument) of the function * is sufficient.Esteva amp; Godo (2001) In general t-norm fuzzy logics, therefore, only left-continuity of * is required, which expresses the assumption that a microscopic decrease of the truth degree of a conjunct should not macroscopically decrease the truth degree of conjunction.

These assumptions make the truth function of conjunction a left-continuous t-norm, which explains the name of the family of fuzzy logics (t-norm based). Particular logics of the family can make further assumptions about the behavior of conjunction (for example, Intermediate logic|Gödel logic requires its idempotence) or other connectives (for example, the logic IMTL requires the involution (mathematics)|involutiveness of negation).

The latter can be interpreted as a fuzzy version of the modus ponens rule of inference. The residuum of a left-continuous t-norm thus can be characterized as the weakest function that makes the fuzzy modus ponens valid, which makes it a suitable truth function for implication in fuzzy logic. Left-continuity of the t-norm is the necessary and sufficient condition for this relationship between a t-norm conjunction and its residual implication to hold.

The set of all *\mboxtautologies is called the logic of the t-norm *, as these formulae represent the laws of fuzzy logic (determined by the t-norm) which hold (to degree 1) regardless of the truth degrees of atomic formulae

* Product fuzzy logic is the logic of the T-norm#Prominent examples|product t-norm x*y = x\cdot y

* Basic fuzzy logic BL is the logic of (the class of) all continuous t-norms

T-norm fuzzy logics - History
Some particular t-norm fuzzy logics have been introduced and investigated long before the family was recognized (even before the notions of fuzzy logic or t-norm emerged):

The book also started the investigation of fuzzy logics as non-classical logics with Hilbert-style calculi, algebraic semantics, and metamathematical properties known from other logics (completeness theorems, deduction theorems, complexity, etc.).

Some of the most important t-norm fuzzy logics were introduced in 2001, by Esteva and Godo (monoidal t-norm logic|MTL, IMTL, SMTL, NM, WNM), Esteva, Godo, and Montagna (propositional ŁΠ),Esteva F., Godo L., Montagna F., 2001, The ŁΠ and ŁΠ½ logics: Two complete fuzzy systems joining Łukasiewicz and product logics, Archive for Mathematical Logic '40': 39–67

T-norm fuzzy logics - Logical language
The logical vocabulary of propositional logic|propositional t-norm fuzzy logics standardly comprises the following connectives:

* 'Implication' \rightarrow (arity|binary). In the context of other than t-norm-based fuzzy logics, the t-norm-based implication is sometimes called 'residual implication' or 'R-implication', as its standard semantics is the t-norm#Residuum|residuum of the t-norm that realizes strong conjunction.

* 'Bottom' \bot (nullary); 0 or \overline are common alternative signs and 'zero' a common alternative name for the propositional constant (as the constants bottom and zero of substructural logics coincide in t-norm fuzzy logics). The proposition \bot represents the falsity or absurdum and corresponds to the classical truth value false.

* 'Top' \top (nullary), also called 'one' and denoted by 1 or \overline (as the constants top and zero of substructural logics coincide in t-norm fuzzy logics). The proposition \top corresponds to the classical truth value true and can in t-norm logics be defined as

* 'Strong disjunction' \oplus (binary). In the context of substructural logics it is also called group, intensional, multiplicative, or parallel disjunction. Even though standard in contraction-free substructural logics, in t-norm fuzzy logics it is usually used only in the presence of involutive negation, which makes it definable (and so axiomatizable) by de Morgan's law from strong conjunction:

T-norm fuzzy logics - Semantics
Algebraic semantics (mathematical logic)|Algebraic semantics is predominantly used for propositional t-norm fuzzy logics, with three main classes of algebraic structure|algebras with respect to which a t-norm fuzzy logic L is completeness (logic)|complete:

T-norm fuzzy logics - Bibliography
* Gottwald S. Hájek P., 2005, Triangular norm based mathematical fuzzy logic. In E.P. Klement R. Mesiar (eds.), Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms, pp.275–300. Elsevier, Amsterdam 2005.

T-norm fuzzy logics - Bibliography
* Hájek P., 1998, Metamathematics of Fuzzy Logic. Dordrecht: Kluwer. ISBN

Instructional Simulation - Hermeneutics, fuzzy logic, and chaos theory
Finally, fuzzy logic is based on the idea that reality is rarely bivalent, but rather multivalent – in other words, there are many in-between values that need to be designed for

Law of thought - Fuzzy Logic
'Fuzzy logic' is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact.

Type-2 fuzzy sets and systems - Interval Type-2 Fuzzy Logic Systems
This kind of FLS is used in fuzzy logic control, fuzzy logic signal processing, rule-based classification, etc., and is sometimes referred to as a function approximation application of fuzzy sets, because the FLS is designed to minimize an error function.

Trust metric - Fuzzy logic
it has been demonstrated that fuzzy logic allows to solve security issues in reliable and efficient manner.

Super Furry Animals - 1996–1998: Fuzzy Logic to Out Spaced
In May, their debut album Fuzzy Logic (album)|Fuzzy Logic was released, again to wide critical acclaim

Super Furry Animals - 1996–1998: Fuzzy Logic to Out Spaced
The reviews were, if anything, better than those for Fuzzy Logic, and it sold more quickly than its predecessor, reaching a peak of No.8: however, Creation did not serve the album particularly well by releasing it just four days after the long-awaited new effort from Oasis, Be Here Now (album)|Be Here Now

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