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Fuzzy Logical Algebras and Their Applications The Scientific World Journal Fuzzy Logical Algebras and Their Applications Guest Editors: Jianming Zhan, Bijan Davvaz, Wieslaw A. Dudek, Young Bae Jun, and Hee Sik Kim Hindawi Publishing Corporation e Scientific World Journal Volume 2015, Article ID 682648, 2 pages http://dx.doi.org/10.1155/2015/682648 Editorial Fuzzy Logical Algebras and Their Applications Jianming Zhan,1 Bijan Davvaz,2 Wieslaw A. Dudek,3 Young Bae Jun,4 andHeeSikKim5 1 Department of Mathematics, Hubei University for Nationalities, Enshi, Hubei 445000, China 2Department of Mathematics, Yazd University, P.O. Box 89195-741, Yazd, Iran 3Institute of Mathematics and Computer Science, Wroclaw University of Technology, 50-370 Wroclaw, Poland 4Department of Mathematics Education, Gyeongsang National University, Jinju 660-701, Republic of Korea 5Department of Mathematics, Hanyang University, Seoul 133-791, Republic of Korea Correspondence should be addressed to Jianming Zhan; [email protected] Received 28 January 2015; Accepted 28 January 2015 Copyright © 2015 Jianming Zhan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. It is well known that an important task of the artificial intel- An important class of algebras inspired by logic is BL- ligenceistomakeacomputersimulateahumanbeingin algebras introduced by Hajek in order to provide an algebraic dealing with certainty and uncertainty in information. Logic proofofthecompletenesstheoremofBasicLogic.Aclassical gives a technique for laying the foundations of this task. example of a BL-algebra is the interval [0, 1] endowed with Information processing dealing with certain information is the structure induced by a continuous t-norm. Other impor- based on the classical logic. Nonclassical logic including tant examples of BL-algebras are MV-algebras introduced many-valued logic and fuzzy logic takes the advantage of the earlier as a model of some infinitely valued Łukasiewicz logic. classical logic to handle information with various facets of All these algebras are also strongly connected with residuated uncertainty, such as fuzziness and randomness. Therefore, lattices. Hence, the filter theory plays an important role in nonclassical logic has become a formal and useful tool for the study of these algebras. From logic point of view, various computer science to deal with fuzzy information and uncer- filters correspond to various sets of provable formulae in the tain information. respective logic. For investigation of several properties, such logics have Belowwepresentseveralpapersonfiltersinvariousalge- been represented as algebras, that is, sets with one, two, bras. In one of these papers, it is proved that there is a one-to- or more algebraic operations satisfying some conditions one correspondence between the set of all vague filters and all inspired by these logics. This inspiration is illustrated by the vaguecongruenceofalatticeimplicationalgebra.Inanother, similarities between the names. We have BCK-algebras and the relations among fuzzy t-filters on residuated lattices are a BCK positive logic, BCI-algebras and a BCI positive logic, described. Another paper is devoted to the filter topology on and BL-algebras and a basic logic. lattice implication algebras. In further articles is characterized In many cases, the connection between such algebras and the relationship between the main types of fuzzy filters in BE- their corresponding logics is much stronger. In this case, one algebras and EQ-algebras. The role of ideals and soft sets in can give a translation procedure which translates all well- BL-algebras is described in another two papers. formed formulas and all theorems of a given logic L into In papers on pseudo-weak-R0 algebras, the most sim- terms and theorems of the corresponding algebras. In some plified axioms system is presented and it is proved that cases, one can give also an inverse translation. In this case, pseudo-weak-R0 algebras are categorically isomorphic to we say that the given logic and class of such algebras are pseudo-IMTL algebras. Pseudo-R0 algebras are categorically isomorphic. Nevertheless, the study of algebras motivated by isomorphic to pseudo-NM algebras. logics is interesting and very useful also in the case when these Several presented papers are devoted to the possible structures are not isomorphic. applications of algebras inspired by logic and fuzzy sets. 2 The Scientific World Journal For example, a robust fuzzy logic controller is proposed for stabilization and disturbance rejection in nonlinear control systems of a particular type. The dynamic feedback con- troller is designed as a combination of a control law that compensates for nonlinear terms in a control system and a dynamic fuzzy logic controller that addresses unknown model uncertainties and an unmeasured disturbance. A mathematical derivation is used to prove that the controller is able to achieve asymptotic stability by processing state measurements. Properties of some hesitant triangular fuzzy aggregation operators based on Bonferroni means also are discussed. In one paper, the method of a construction of a ranking of the physical matches is proposed. This method is basedonthefuzzyclusteringanalysis. Decisions are often made by many decision experts having preference of risks. In view of the grey situation group decision-making problems, a grey situation group decision-making method on the basis of prospect theory is shown. The method takes the positive and negative ideal situation distance as reference points, defines positive and negative prospect value function, and introduces decision experts’ risk preference into grey situation decision-making to make the final decision more in line with decision experts’ psychological behavior. One possibility of determination of the weight of each decision expert sets up comprehensive prospect value matrix for decision experts’ evaluation, and finally determination of the optimal situation is presented. Jianming Zhan Bijan Davvaz Wieslaw A. Dudek Young Bae Jun Hee Sik Kim Hindawi Publishing Corporation e Scientific World Journal Volume 2015, Article ID 682989, 11 pages http://dx.doi.org/10.1155/2015/682989 Research Article A New Fuzzy System Based on Rectangular Pyramid Mingzuo Jiang,1 Xuehai Yuan,1,2 Hongxing Li,1 and Jiaxia Wang3 1 Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116024, China 2Foundation Department, Dalian University of Technology, Panjin 124221, China 3Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116024, China Correspondence should be addressed to Xuehai Yuan; [email protected] Received 10 July 2014; Accepted 7 September 2014 Academic Editor: Jianming Zhan Copyright © 2015 Mingzuo Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A new fuzzy system is proposed in this paper. The novelty of the proposed system is mainly in the compound of the antecedents, which is based on the proposed rectangular pyramid membership function instead of t-norm. It is proved that the system is capable of approximating any continuous function of two variables to arbitrary degree on a compact domain. Moreover, this paper provides one sufficient condition of approximating function so that the new fuzzy system can approximate any continuous function of two variables with bounded partial derivatives. Finally, simulation examples are given to show how the proposed fuzzy system can be effectively used for function approximation. 1. Introduction on the structure design of conventional dual-input single- output Mamdani fuzzy system to improve its approximation Fuzzy system has been the subject of numerous researches property and simplify its structure. A new kind of fuzzy sys- in more than three decades. It has been successfully applied tem based on the proposed rectangular pyramid membership to a great variety of different processes such as control engi- function is established. The model of the rectangular pyramid neering, signal processing, information processing, machine fuzzy system (RPFS) is introduced mainly by replacing the intelligence, decision making, management, finance, medi- compound of the two rule antecedents using t-norm with cine, and robotics [1–9]. the rectangular pyramid membership function of the input Motivated by successful applications of fuzzy system, vector. With the help of rectangular pyramid membership there have been a number of works aiming at improving function, the fuzzy system structure becomes simple and thestructureandperformanceoffuzzysystem.Yuanetal. easily realized. [10] put forward a parameter singleton fuzzifier method. In most applications of fuzzy systems, the main design Celikyilmaz and Turksen [11] presented a new fuzzy system objective can be considered as problems of functions approx- modeling approach based on improved fuzzy functions to imation. So the study on approximation theory of fuzzy sys- model systems with continuous output variable. Adaptive tems is very important and necessary. Wang [27]usedStone- fuzzy systems are also developed well. In [12], a dynamic Weierstrass Theorem to prove the approximation capability rule base which allows the fuzzy sets to dynamically change of a common
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