Fuzzy Logic

Overview The reasoning in brumous logic is similar to human reasoning. It allows for approximate values and inferences as well as incomplete or ambiguous data ( blear data) as opposed to only relying on crisp data (binary yes/no choices). Fuzzy logic is able to impact incomplete data and provide approximate solutions to problems other methods find difficult to solve. Terminology used in clouded logic non used in other methods be very high, increasing, somewhat decreased, reasonable and very low. 4 editDegrees of rectitudeFuzzy logic and probabilistic logic atomic number 18 numerally similar both(prenominal) be possessed of truth values ranging between 0 and 1 but conceptually distinct, due to polar interpretationssee interpretations of chance theory. Fuzzy logic corresponds to degrees of truth, while probabilistic logic corresponds to probability, likelihood as these differ, muzzy logic and probabilistic logic yield different models of the same real-world situations. Both degrees of truth and probabilities verify 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 meaning of each of them passel be represented by a certain fuzzy post. Then one might define the glass as being 0. 7 empty and 0. 3 full. Note that the concept of emptiness would be inwrought and thus would depend on the observer or designer. Another designer might equally well design a coif membership make for where the glass would be considered full for all values down to 50 ml. 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. editApplying truth values A base application might characterize subranges of a sustained variable. For instance, a temperature measurement for anti-lock brakes might have some(prenominal) separate membership personas defini ng particular temperature ranges infallible 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 influence how the brakes should be controlled. Fuzzy logic temperature In this image, the meaning of the expressions cold, warm, and hot is represented by functions mapping a temperature outdo.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 pointers (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 slightly warm and the olive-drab arrow (pointing at 0. 8) fairly cold. editLinguistic 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. 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 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- accordingly rules, or constructs that are equivalent, such as fuzzy associative matrices.Rules are usually expressed in the represent 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 There is no ELSE all of the rules are evaluated, because the te mperature might be cold and normal at the same time to different degrees. The AND, OR, and not 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)) There are also 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. 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 essential propositional fuzzy logics are Monoidal t-norm-based propositional fuzzy logic MTL is an maximatization of logic where conjunction is defined by a left perpetual t-norm, and implication is defined as the equaliser 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 generation 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 harvesting t-norm.It has the axioms of BL plus another axiom for cancellativity of conjunction, a nd 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 traditional 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 persist 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. editDecidability issues for fuzzy logic The notions of a decidable subset and recursively enumerable subs et 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 definition is not adequate and therefore proposed the following one. U denotes the set of thinking(prenominal) 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 recursively 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 related with fuzzy logic. Indeed, the following theorem holds sure (provide d 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 align 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 hand over 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 open uestion is to start from this notion to find an extension of Godels theorems to fuzzy logic. editFuzzy databases Once fuzzy relations are defined, it is thinkable to develop fuzzy relational databases. The first fuzzy relational database, F RDB, 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 acknowledge 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 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 alternate 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 dyed intelligence Artificial neural network Defuzzification Dynamic logic Expert sys tem False dilemma Fuzzy architectural spatial analysis Fuzzy associative matrix Fuzzy classificationFuzzy concept Fuzzy Control Language Fuzzy Control System Fuzzy electronics Fuzzy mathematics Fuzzy set Fuzzy subalgebra FuzzyCLIPS expert system Machine learning Multi-valued logic Neuro-fuzzy Paradox of the heap Rough set Type-2 fuzzy sets and systems Vagueness Interval finite particle 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. ). bleak York McGraw Hill. pp. 431. Zadeh, L. 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