Fuzzy Logic

Overview The reasoning in blear remains of system of logical systemal system is similar to human reasoning. It allows for approximate determine and inferences as well as incomp permite or ambiguous data ( brumous data) as opposed to yet relying on crisp data (binary yes/no choices). hirsute logic is fitted to process incomplete data and provide approximate solutions to troubles former(a) methods regulate difficult to solve. Terminology used in logy logic non used in other methods ar very high, increasing, somewhat decreased, sane and very low. 4 editDegrees of law fuzzed logic and probabilistic logic be numerally similar both stick law mints ranging amongst 0 and 1 but imageually distinct, due to different interpretations detect interpretations of opportunity surmisal. hairy logic corresponds to degrees of truth, while probabilistic logic corresponds to probability, likelihood as these differ, woolly logic and probabilistic logic break different m odels of the same real-world situations. Both degrees of truth and probabilities range between 0 and 1 and hence may seem similar at first. For example, let a 100 ml glass contain 30 ml of water.Then we may consider two concepts Empty and Full. The content of severally of them can be equal by a certain(p) hirsute desexualise. Then bingle might define the glass as beingness 0. 7 empty and 0. 3 full. Note that the concept of emptiness would be subjective and thus would depend on the observer or designer. other designer might equally well design a set membership function where the glass would be conside inflamed full for all values elaborate to 50 ml. It is essential to realize that blear logic uses truth degrees as a mathematical model of the vagueness phenomenon while probability is a mathematical model of ignorance. editApplying truth values A basic practise might characterize subranges of a continuous variable. For instance, a temperature measurement for anti-lock stop might have several separate membership functions defining particular(prenominal) temperature ranges needed to control the brake system 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. fogged logic temperature In this image, the meaning of the fonts dusty, warm, and eager is represented by functions mapping a temperature home plate.A point on that scale has three truth valuesone 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. The orange arrow (pointing at 0. 2) may describe it as about warm and the blue arrow (pointing at 0. 8) fairly cold. editLinguistic variables man variables in maths usually take numerical values, in groggy logic applications, the non-numeric li nguistic variables be often used to facilitate the expression of rules and facts. 5 A linguistic variable such as age may have a value such as young or its antonym old. However, the great utility of linguistic variables is that they can be modify via linguistic hedges applied to primary terms. The linguistic hedges can be associated with certain functions. editExample Fuzzy set theory defines addled operators on blear sets. The problem in applying this is that the appropriate bleary operator may not be known. For this reason, befuddled logic usually uses IF- wherefore rules, or constructs that are equivalent, such as blear-eyed associatory matrices.Rules are usually expressed in the image IF variable IS property THEN action For example, a simple temperature regulator that uses a fan might look like this IF temperature IS very cold THEN stop fan IF temperature IS cold THEN turn down fan IF temperature IS normal THEN maintain level IF temperature IS hot THEN speed up fan Ther e is no ELSE all of the rules are evaluated, because the temperature might be cold and normal at the same eon to different degrees. The AND, OR, and NOT operators of boolean logic exist in blurry logic, usually delineate as the minimum, maximum, and omplement when they are defined this way, they are called the Zadeh operators. So for the woolly variables x and y NOT x = (1 truth(x)) x AND y = minimum(truth(x), truth(y)) x OR y = maximum(truth(x), truth(y)) There are besides other operators, more linguistic in nature, called hedges that can be applied. These are by and large adverbs such as very, or somewhat, which modify the meaning of a set using a mathematical formula. editLogical depth psychology In mathematical logic, there are several formal systems of fuzzy logic most of them belong among so-called t-norm fuzzy logics. editPropositional fuzzy logics The most central propositional fuzzy logics are Monoidal t-norm-based propositional fuzzy logic MTL is an maximatizatio n of logic where conjunction is defined by a left continuous t-norm, and implication is defined as the balance wheel of the t-norm. Its models correspond to MTL-algebras that are prelinear commutative bounded integral residuated lattices. staple 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. Lukasiewicz fuzzy logic is the extension of basic fuzzy logic BL where standard conjunction is the Lukasiewicz t-norm. It has the axioms of basic fuzzy logic plus an axiom of double negation, and its models correspond to MV-algebras. Godel fuzzy logic is the extension of basic fuzzy logic BL where conjunction is Godel 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 harvest-time t-norm.I t 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 Pavelkas logic), denoted by EVL, is a further generalization of mathematical fuzzy logic. While the above kinds of fuzzy logic have conventional syntax and many-valued semantics, in EVL is evaluated also syntax. This means that each formula has an evaluation. Axiomatization of EVL stems from Lukasziewicz fuzzy logic. A generalization of classical Godel completeness theorem is provable in EVL. editPredicate fuzzy logics These tense up the above-mentioned fuzzy logics by adding universal and existential logical 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. editDecidability issues for fuzzy logic The notions of a decidable subset and recursively calculable subset are basic ones for classical mathematics and classical logic.Then, the question of a suitable extension of such concepts to fuzzy set theory arises. A first proposal in such a direction was make by E. S. Santos by the notions of fuzzy Turing machine, Markov normal fuzzy algorithm and fuzzy program (see Santos 1970). Successively, L. Biacino and G. Gerla showed that such a definition is not adequate and hence proposed the following one. U denotes the set of rational numbers in 0,1. A fuzzy subset s S 0,1 of a set S is recursively enumerable if a recursive map h S?N U exists such that, for every x in S, the function h(x,n) is increasing with jimmy to n and s(x) = lim h(x,n). 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 chemise of the L-subsets is proposed in Gerla 2006. The proposed definitions are well related with fuzzy logic. Indeed, the following theorem holds current (provided that the deduction apparatus of the fuzzy logic satisfies some obvious forcefulness property). Theorem. 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 up to(p) question to give supports for a Church thesis for fuzzy logic claiming that the proposed notion of recursive enumerability for fuzzy subsets is the adequate one. To this aim, further investigations on the notions of fuzzy grammar and fuzzy Turing machine should be necessary (see for example Wiedermanns paper). Another disperse uestion is to start from this notion to find an extension of Godels theorems to fuzzy logic. editFuzzy databases at a time fuzzy relations are defined, it is p ossible to develop fuzzy relative databases. The first fuzzy relational database, FRDB, appeared in Maria Zemankovas 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. M. Medina, M. A. Vila et al. In the context of fuzzy databases, some fuzzy querying languages have been defined, highlighting 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 and so on. editComparison to probability Fuzzy logic and probability are different ways of expressing uncertainty. While both fuzzy logic and probability theory can be used to represent subjective belief, fuzzy set theory uses the concept of fuzzy set membership (i. e. , how oft a variable is in a set), and probability theory use s the concept of subjective probability (i. . , how probable do I think that a variable is in a set). While this distinction is mostly philosophical, the fuzzy-logic-derived disaster measure is inherently different from the probability measure, hence they are not directly equivalent. However, many statisticians are persuaded by the work of Bruno de Finetti that only one kind of mathematical uncertainty is needed and thus fuzzy logic is unnecessary. On the other hand, baronet Kosko arguescitation needed that probability is a subtheory of fuzzy logic, as probability only handles one kind of uncertainty.He also claimscitation needed to have proven a derivation of Bayes theorem from the concept of fuzzy subsethood. Lotfi 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 what is called possibility theory. (cf. 6) editSee also Logic portal Thinking portal fak e intelligence Artificial neural network Defuzzification Dynamic logic Expert system ill-judged dilemma Fuzzy architectural spatial analysis Fuzzy associative matrix Fuzzy classificationFuzzy concept Fuzzy guarantee Language Fuzzy Control System Fuzzy electronics Fuzzy mathematics Fuzzy set Fuzzy subalgebra FuzzyCLIPS expert system Machine encyclopedism Multi-valued logic Neuro-fuzzy Paradox of the heap Rough set Type-2 fuzzy sets and systems Vagueness Interval finite element Noise-based logic editNotes Novak, V. , Perfilieva, I. and Mockor, J. (1999) mathematical principles of fuzzy logic Dodrecht Kluwer Academic. ISBN 0-7923-8595-0 Fuzzy Logic. Stanford Encyclopedia of Philosophy. Stanford University. 2006-07-23. Retrieved 2008-09-29. Zadeh, L. A. (1965). Fuzzy sets, nurture and Control 8 (3) 338353. James A. OBrien George M. Marakas (2011). Management Information Systesm (10th ed. ). New York McGraw Hill. pp. 431. Zadeh, L. 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