Fuzzy Logic

Overview
The reasoning in fuzzy logic is similar to human reasoning. It allows for approximate values and inferences as well as incomplete or ambiguous data (fuzzy data) as opposed to only relying on crisp data (binary yes/no choices). Fuzzy logic is able to process incomplete data and provide approximate solutions to problems other methods find difficult to solve. The terminology used in fuzzy logic not used in other methods is: very high, increasing, somewhat decreased, reasonable, and very low. [4]  Fuzzy logic and probabilistic logic are mathematically similar – both have truth values ranging between 0 and 1 – but conceptually distinct, due to different interpretations—see interpretations of probability theory. Fuzzy logic corresponds to “degrees of truth”, while probabilistic logic corresponds to “probability, likelihood”; as these differ, fuzzy logic and probabilistic logic yield different models 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 meaning of each of them can be represented by a certain fuzzy set. Then one might define the glass as being 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. 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 realize that fuzzy logic uses truth degrees as a mathematical model of the vagueness phenomenon while probability is a mathematical model of ignorance. edit]Applying truth values A basic application might characterize subranges of a continuous variable. For instance, a temperature measurement for anti-lock brakes might have several separate membership functions defining particular temperature ranges needed 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 determine 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 scale.

A point on that scale has three “truth values”—one 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 “slightly warm” and the blue arrow (pointing at 0. 8) “fairly cold”. 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. [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 modified via linguistic hedges applied to primary terms. The linguistic hedges can be associated with certain functions. [edit]Example 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.
Rules are usually expressed in the form: 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 temperature 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 complement; 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. [edit]Logical analysis In mathematical logic, there are several formal systems of “fuzzy logic”; most of them belong among so-called t-norm fuzzy logics. edit]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 the implication is defined as the residuum of the t-norm. Its models correspond to MTL-algebras that are pre-linear 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 the 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 product t-norm. It has the axioms of BL plus another axiom for cancelability of conjunction, and its models are called product algebras. Fuzzy logic with evaluated syntax (sometimes also called Pavelka’s 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’s fuzzy logic. A generalization of the classical Godel completeness theorem is provable in EVL. Predicate fuzzy logic These extend the above-mentioned fuzzy logic 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. [edit]Decidability issues for fuzzy logic The notions of a “decidable subset” and “recursively enumerable subset” are basic ones for classical mathematics and classical logic.
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 to 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 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 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 the fuzzy Turing machines should be necessary (see for example Wiedermann’s paper). Another open question is to start from this notion to find an extension of Godel’s theorems to fuzzy logic. Fuzzy databases 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. 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. Comparison 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, the 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 argues[citation needed] that probability is a sub theory of fuzzy logic, as probability only handles one kind of uncertainty. He also claims[citation 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. See also Logic portal Thinking portal Artificial intelligence Artificial neural network Defuzzification Dynamic logic Expert system False dilemma Fuzzy architectural spatial analysis Fuzzy associative matrix Fuzzy classification
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