Fuzzy logic - Scholarpedia. Humans have a remarkable capability to reason and make decisions in an environment of uncertainty, imprecision, incompleteness of information, and partiality of knowledge, truth and class membership. The principal objective of fuzzy logic is formalization/mechanization of this capability. This video introduces fuzzy logic, including the basics of fuzzy sets, fuzzy rules and how these are combined in decision making.The core of FL is Graduation/Granulation, G/G. There are many misconceptions about fuzzy logic. To begin with, fuzzy logic is not fuzzy. In large measure, fuzzy logic is precise. Another source of confusion is the duality of meaning of fuzzy logic. In a narrow sense, fuzzy logic is a logical system. But in much broader sense which is in dominant use today, fuzzy logic, or FL for short, is much more than a logical system. Fuzzy sets, fuzzy logic, and fuzzy control systems A type of logic that recognizes more than simple true and false values. With fuzzy logic, propositions can be represented with degrees of truthfulness and falsehood. For example, the statement, today is sunny,might be 100% true if there are no clouds, 80% true if there are a few clouds, 50% true if. More specifically, fuzzy logic has many facets (Figure 1). There are four principal facets. The fuzzy- set- theoretic facet, FLs. The logical facet, FLl. The product guides you through the steps of designing fuzzy inference systems. Functions are provided for many common methods, including fuzzy clustering. This definition explains what fuzzy logic is and how it's used in computing and data analytics applications. See also: A discussion of fuzzy logic's history and inks to more. The epistemic facet, FLe; and. The relational facet, FLr. More specifically, in fuzzy logic everything is or is allowed to be graduated, that is, be a matter of degree or, equivalently, fuzzy. Furthermore, in fuzzy logic everything is or is allowed to be granulated, with a granule being a clump of attribute- values drawn together by indistinguishability, similarity, proximity or functionality. For example, Age is granulated when its values are described as young, middle- aged and old (Figure 2). A linguistic variable may be viewed as a granulated variable whose granular values are linguistic labels of granules. In a qualitative way, graduation and granulation play pivotal roles in human cognition. The concept of a fuzzy set was introduced in (Zadeh 1. The theory of fuzzy sets is central to fuzzy logic (Pedrycz and Gomide 1. A fuzzy set, A, in a universe of discourse, U, is defined by a membership function which associates with each object, u, in U, the degree to which u is a member of A. A fuzzy set is basic if its membership function takes values in the unit interval. More generally, the membership function may take values in a partially ordered set. There are many types of fuzzy sets, among them fuzzy sets of Type 2 (Zadeh 1. Mendel 2. 00. 0), L- Fuzzy sets (Goguen 1. Zhang 1. 99. 8; Benferhat, Dubois, Kaci and Prade 2. Atanassov 1. 98. 6). In a general setting, intersection and union of fuzzy sets are defined in terms of t- norms and t- conorms (Klement, Mesiar and Pap 2. FLl may be viewed as a generalization of multivalued logic. The agenda of FLl is similar in spirit to the agenda of classical logic (Hajek 1. Novak, Perfilieva and Mockor 1. Truth values in FLl are allowed to be fuzzy sets. In FLe, a natural language is viewed as a system for describing perceptions. An important branch of FLe is possibility theory (Zadeh 1. Dubois and Prade 1. Another important branch of FLe is the computational theory of perceptions (Zadeh 1. In FLr, a granulated function, \(f*\ ,\) is described as a collection of fuzzy if- then rules of the form: if \(X\) is \(A\) then \(Y\) is \(B\ ,\) where \(A\) and \(B\) are fuzzy sets carrying linguistic labels like small, medium, and large (Figure 3). In this sense, \(X\) and \(Y\) are linguistic variables (Zadeh 1. The concept of a linguistic variable and the associated calculi of fuzzy if- then rules (Zadeh 1. Mamdani and Assilian 1. Bardossy and Duckstein 1. A granulated function, \(f*\ ,\) may be viewed as a summary of \(f\ ,\) with \(f*\) being a granular value of \(f\ .\) An important special case of a granular function is a granular probability density function (Figure 4). In this perspective, perception of a probability distribution may be described as a granular probability distribution. Constraints are ubiquitous. In scientific theories, representation of constraints is generally oversimplified. Oversimplification of constraints is a necessity because existing constraint definition languages have a very limited expressive power. The concept of a generalized constraint is intended to provide a basis for construction of a maximally expressive constraint definition language- -a language which can also serve as a meaning representation/precisiation language for natural languages. The principal modalities are. The standard constraints are bivalent possibilistic, bivalent veristic and probabilistic. Standard constraints have a position of centrality in existing scientific theories. A generalized constraint, GC(\(X\)), is open if \(X\) is a free variable, and is closed if \(X\) is instantiated. A proposition is a closed generalized constraint. Unless indicated to the contrary, a generalized constraint is assumed to be closed. A generalized constraint may be generated by combining, projecting, qualifying, propagating and counterpropagating other generalized constraints. The set of all generalized constraints together with the rules governing combination, projection, qualification, propagation and counterpropagation, constitutes the Generalized Constraint Language (GCL). More specifically, a key idea in fuzzy logic is that of representing the information about a variable \(X\ ,\) \(I(X)\ ,\) as a generalized constraint on \(X\ ,\) GC(\(X\)). The symbolic equation \(I(X)\)=GC(\(X\)) is the fundamental thesis of fuzzy logic. A consequence of the fundamental thesis is that the meaning of a proposition, \(p\ ,\) is expressible as a generalized constraint. This is the meaning postulate of fuzzy logic. More specifically, the meaning of a proposition is expressible as a closed generalized constraint, while the meaning of a predicate is expressible as an open generalized constraint. Equivalently, if \(p\) is a proposition or a predicate then the meaning postulate may be stated as an assertion that the meaning of \(p\) may be represented/precisiated through translation of \(p\) into GCL. A very simple example of annotated translation is. Lily is young \(X\)/Age(Lily) is \(R\)/young. X\) is Age(Lily), the constraining relation, \(R\ ,\) is young, and the constraint is possibilistic (\(r\)=blank). Equivalently, the meaning postulate implies that the meaning of a proposition or a predicate is defined by identifying the constrained variable, \(X\ ,\) the constraining relation, \(R\ ,\) and the modality of the constraint, \(r\ .\). Granular computing. There is a close connection between the concept of a generalized constraint and the concept of a granular value. More specifically, if \(X\) is a variable taking values in a universe of discourse, \(U\ ,\) then \(a\) is a singular value of \(X\) if \(a\) is a singleton, implying that there is no uncertainty or imprecision about the value of \(X\ .\) If this is not the case, then a granular value of \(X\ ,\) \(A\ ,\) may be viewed as a representation of the state of knowledge about the value of \(X\) (Figure 5). For example, if \(X\) is unemployment, then 9. X\ ,\) and . Symbolically, \(A\)=GC(\(X\)). In the unemployment example, . In granular computing, the objects of computation are granular values which are defined as generalized constraints. Granular computing is rooted in (Zadeh 1. The term Granular Computing was suggested by T. Y. Pedrycz is the first book on granular computing (Bargiela and Pedrycz, 2. Since a natural language is a system for describing perceptions, NL- Computation is closely related to computation with perception- based information. As an illustration, if my perception is that most Swedes are tall, then what is the average height of Swedes? Another example: Robert usually leaves office at about 6 pm. Usually it takes him about an hour to get home. What is the probability that Robert is home after about 7 pm? NL- capability is the capability of a theory to operate on information described in natural language or, equivalently, to operate on perception- based information. The importance of NL- capability derives from the fact that much of human knowledge is expressed in natural language. In the first stage, the information which is described in a natural language is precisiated through translation into the Generalized Constraint Language. The result is granular information expressed as system of generalized constraints. The second stage involves granular computing. Finally, the result of granular computing is retranslated into natural language. The principal rule is the extension principle. Extension principle has many versions. The simplest version (Zadeh 1. Let \(f\) be a function from reals to reals, \(Y=f(X)\ .\) What we know is that \(X\) is \(A\ ,\) where \(A\) is a fuzzy subset of the real line. Equivalently, what we know about \(X\) is its granular value, that is, its possibility distribution, \(A\ .\) What can be said about \(Y\ ,\) that is, what is its granular value or, equivalently, its possibility distribution? In a more general form, (Zadeh 1. X\) is \(A\) is replaced by \(f(X)\) is \(A\) (Figure 6). It is this form that is used in most practical applications. In a form that is used in fuzzy control, what is granulated is \(f\ ,\) resulting in a granular function, \(f^*\ ,\) which is defined by a collection of fuzzy- if- then rules. Let \(Y=f(X)\ ,\) where \(X\) is a real- valued variable. Assume that we can compute \(Y\) for singular values of \(f\) and \(X\ .\) Basically, the extension principle serves to extend the definition of \(Y\) to granular values of \(f\) and \(X\ .\). History. During much of its early history, fuzzy logic has been an object of skepticism and derision, in part because fuzzy is a word which is usually used in a pejorative sense. Today, fuzzy logic has an extensive literature and a wide variety of applications ranging from consumer products and fuzzy control to medical diagnostic systems and fraud detection (Zadeh 1. Novak and Perfilieva 2. What is widely unrecognized is that many scientific theories can be enriched through addition of concepts and techniques drawn from fuzzy logic. In particular, fuzzy logic can add to existing theories NL- capability, that is, the capability to operate on information described in natural language or, equivalently, on perception- based information.
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