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How to Untangle Complex Systems?
Published in Pier Luigi Gentili, Untangling Complex Systems, 2018
It derives that Fuzzy logic is an infinite-valued logic. Fuzzy logic is suitable to describe any complex non-linear cause and effect relation after the construction of a proper Fuzzy Logic System (FLS). The development of a FLS requires three fundamental steps. First, the granulation of every variable in Fuzzy sets (see Figure 13.17a). In other words, all the possible values of a variable are partitioned in a certain number of Fuzzy sets. The number, position, and shape of the Fuzzy sets are context-dependent. The second step is the graduation of all the variables: each Fuzzy set is labeled by a linguistic variable, often an adjective (see Figure 13.17b as an example for the variable temperature expressed in degree Celsius). These two steps imitate how our senses work (Gentili 2017). Physical and chemical sensations are epitomized by words in natural language. For instance, human perceptions of temperature are described by adjectives such as “freezing,” “cold,” “temperate,” “warm,” “hot,” and so on. Visible radiations of different spectral compositions are described by words such as “green,” “blue,” “yellow,” “red,” and other words that we call colors. Savors are classified as salt, sweet, bitter, acid, umami. There are different words to describe the other sensations. These words of the natural language are cognitive granules originated by perceptions. Perceptions are constrained, noisy, and they generate clumps of Fuzzy-information (remember what we studied in Chapter 12).
Introduction to Expert Systems
Published in Chris Nikolopoulos, Expert Systems, 1997
Fuzzy logic is an attempt to do away with the boolean paradigm and deal with vagueness in the knowledge. Fuzzy set theory and fuzzy logic were developed by Zadeh, ([33]), extending Lukasiewicz's n-valued logics. In n-valued logic, the set of truth values was extended from just 0 or 1 (true or false) to values in the truth set Tn={0,1/(n-1),2/(n-1),…,, (n-2)/(n-1),1}. For example, in 3-valued logic the truth values allowed are 0,1/2 and 1, corresponding to false, unknown and true. The standard Lukasiewicz logic L1, is an infinite valued logic, where the truth values can span the whole interval [0,1] and corresponds to Zadeh's fuzzy set theory.
FUZZY SETS, FUZZY OPERATORS, AND FUZZY RELATIONS
Published in Kumar S. Ray, Soft Computing and Its Applications, Volume One, 2014
The notion of an infinite-valued logic was introduced in Zadeh's seminal work "Fuzzy Sets" where he described the mathematics of fuzzy set theory, and by extension fuzzy logic. This theory proposed making the membership functions linguistic truth values (that is true, false, very true, more or less false, and so on) operate over the range of real numbers [0,1]. New operations for the calculus of logic were proposed, and showed to be in principle at least a generalization of classic logic.
Solving Arbitrary Coupled Trapezoidal Fully Fuzzy Sylvester Matrix Equation with Necessary Arithmetic Multiplication Operations
Published in Fuzzy Information and Engineering, 2022
Ahmed Abdel Aziz Elsayed, Nazihah Ahmad, Ghassan Malkawi
Fuzzy logic has been studied since the 1920s, as infinite valued logic by Lukasiewicz and Tarski [14], and the fuzzy set theory was introduced by Lotfi Zadeh [15] in 1965 while the set theory was developed by Georg Cantor [16]. Fuzzy Relation Equations (FREs) with the max–min composition was first studied by Sanchez [17]. The theory and applications of FREs can be found in Di Nola et al. [18], which indicated that if the solvability of max-continuous t-norm FREs is assumed, then the solution set for the FREs can be fully determined from a unique greatest solution and all minimal solutions, and the number of minimal solutions is always finite. Since then, FREs based on various compositions have been investigated. Some common compositions include max–min [19–24], max-product [25–28], max-Archimedean t-norm [29, 30], u-norm [31], max t-norm [32] and max arithmetic mean [33,34]. The conditions for the existence of a solution to the inverse problem concerned with FRE are investigated in [35], a finite system of FREs with sup-T composition was studied in [36], and a system of FREs was investigated in [37, 38].
Fuzzy intensional semantics
Published in Journal of Applied Non-Classical Logics, 2018
Throughout the paper we shall work over an arbitrary fixed -core fuzzy logic L. Readers unfamiliar with fuzzy logics can equate L with some well-known specimen of the family, for instance the infinite-valued logic of Łukasiewicz or Gödel–Dummett logic of linear Heyting algebras (both with the additional connective . For details on -core fuzzy logics, including all requisite definitions and properties needed in this paper, we refer the reader to Appendix.