Fuzzy Logic

Overview The abstract thought in blear-eyed system of system of system of logical systemal systemal system is similar to human reasoning. It allows for approximate set and inferences as closely as incomplete or ambiguous data ( muddled data) as opposed to only relying on crisp data (binary yes/no choices). blear-eyed logic is able to process incomplete data and countenance approximate solutions to problems other methods find difficult to solve. Terminology apply in fogged logic non used in other methods atomic number 18 very high, increasing, somewhat decreased, reasonable and very low. 4 editDegrees of justice bleary-eyed logic and probabilistic logic atomic number 18 mathematically similar some(prenominal) buzz off fair play set ranging between 0 and 1 but innovationually distinct, due to different interpretationssee interpretations of chance theory. blear-eyed logic corresponds to degrees of truth, while probabilistic logic corresponds to chance, lik elihood as these differ, haired logic and probabilistic logic yield different simulates of the same real-world situations. Both degrees of truth and probabilities range between 0 and 1 and hence whitethorn reckon similar at first. For example, let a 100 ml sparkler contain 30 ml of water.Then we may consider 2 concepts Empty and Full. The meaning of each of them can be represented by a certain misty set. Then virtuoso might touch on the glass as being 0. 7 empty and 0. 3 full. logical argument that the concept of emptiness would be subjective and thus would depend on the observer or designer. Another designer might equally well design a set membership function where the glass would be considered full for all values down to 50 ml. It is essential to pretend that woolly logic uses truth degrees as a mathematical standard of the vagueness phenomenon while probability is a mathematical model of ignorance. editApplying truth values A basic application might characterize subra nges of a regular variable. For instance, a temperature meatireement for anti-lock brakes might have some(prenominal) key membership functions defining particular temperature ranges involve to control the brakes properly. to each one 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. woolly logic temperature In this image, the meaning of the expressions dust-covered, warm, and hot is represented by functions mapping a temperature scale.A head word 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 interpret as not hot. The orange arrow (pointing at 0. 2) may describe it as slightly warm and the blue arrow (pointing at 0. 8) fairly cold. editLinguistic variables trance var iables in mathematics ordinarily study numerical values, in hirsute logic applications, the non-numeric lingual variables are very much used to facilitate the expression of rules and facts. 5 A linguistic variable much(prenominal) 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 modified via linguistic hedges applied to primary terms. The linguistic hedges can be associated with certain functions. editExample woolly set theory defines muddled instruments on wooly-minded sets. The problem in applying this is that the appropriate addled operator may not be known. For this reason, fogged logic usually uses IF-THEN rules, or constructs that are equivalent, such as brumous associative matrices.Rules are usually expressed in the form IF variable IS property THEN satisfy For example, a simple temperature regulator that uses a fan might opine like this IF temperature IS very cold THEN stop fan I F temperature IS cold THEN turn down fan IF temperature IS average THEN support level IF temperature IS hot THEN speed up fan at that place is no ELSE all of the rules are evaluated, because the temperature might be cold and normal at the same time to different degrees. The AND, OR, and non operators of boolean logic exist in fuzzy logic, usually defined as the minimum, maximum, and omplement when they are defined this way, they are called the Zadeh operators. So for the fuzzy 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)) in that location are withal 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. edit logical analysis In mathematical logic, there are some(prenominal) formal systems of fuzzy logic most of them belong among so-called t-norm fuzzy logics. editPr opositional fuzzy logics The most important propositional fuzzy logics are Monoidal t-norm-based propositional fuzzy logic MTL is an mottoatization of logic where conjunction is defined by a left perpetual t-norm, and implication is defined as the residuum of the t-norm. Its models correspond to MTL-algebras that are prelinear independent bounded integral residuated lattices. Basic propositional fuzzy logic BL is an appendix of MTL logic where conjunction is defined by a continuous t-norm, and implication is withal defined as the residuum of the t-norm.Its models correspond to BL-algebras. Lukasiewicz fuzzy logic is the fender 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 product t-norm.It has the axioms of BL plus another axiom for cancellativity of conjunction, and its models are called product algebras. wooly-minded logic with evaluated syntax (sometimes also called Pavelkas logic), denoted by EVL, is a further generalization of mathematical fuzzy logic. While the above good-natureds of fuzzy logic have traditional syntax and mevery-valued semantics, in EVL is evaluated also syntax. This mode that each formula has an evaluation. Axiomatization of EVL stems from Lukasziewicz fuzzy logic. A generalization of classical Godel completeness theorem is demonstrable in EVL. editPredicate fuzzy logics These extend the above-mentioned fuzzy logics by adding worldwide and existential quantifiers in a manner similar to the way that depose logic is created from propositional logic. The semantics of the universal (resp. existential) quant ifier 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 enumerable 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 made 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 interpretation is not able and therefore proposed the following one. U denotes the set of acute 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 respect to n and s(x) = lim h(x,n). We say that s is decidable if both s and its complement s are recursiv ely enumerable. An extension of such a theory to the general case of the L-subsets is proposed in Gerla 2006. The proposed definitions are well cerebrate with fuzzy logic. Indeed, the following theorem holds true (provided that the deduction apparatus of the fuzzy logic satisfies some obvious effectiveness property). Theorem. Any axiomatizable fuzzy theory is recursively enumerable.In particular, the fuzzy set of logically true formulas is recursively enumerable in ache 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 building 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 needful (see for example Wiedermanns paper). Another open uestion is to start from this notion to find an extension of Godels theorems to fuzzy logic. edit blear-eyed databases Once fuzzy relations are defined, it is assertable to develop fuzzy relational 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 blear-eyed logic and probability are different ways of expressing uncertainty. While both fuzzy logic and probability theory can be used to repre sent subjective belief, fuzzy set theory uses the concept of fuzzy set membership (i. e. , how much a variable is in a set), and probability theory uses 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 possibility 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, Bart 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 site Thinking portal Artificial intelligence Artificial neural network Defuzzification Dynamic logic Expert system False dilemma Fuzzy architectural spatial analysis Fuzzy associative matrix Fuzzy classificationFuzzy concept Fuzzy Control Language Fuzzy Control dust Fuzzy electronics Fuzzy mathematics Fuzzy set Fuzzy subalgebra FuzzyCLIPS safe system Machine learning Multi-valued logic Neuro-fuzzy Paradox of the multitude 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, Information and Control 8 (3) 338353. James A. OBrien George M. Marakas (2011). Management Information Systesm (10th ed. ). brand-new York McGraw Hill. pp. 431. 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