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Journal ArticleDOI

System of fuzzy relation equations as a continuous model of IF-THEN rules

20 Aug 2007-Information Sciences (Elsevier)-Vol. 177, Iss: 16, pp 3218-3227
TL;DR: This paper proposes several new necessary and sufficient criteria for the so-called Mamdani relation to be a solution to the system and proves that in the general case, these same criteria are sufficient (but not always necessary) for the solvability of the system.
About: This article is published in Information Sciences.The article was published on 2007-08-20. It has received 72 citations till now. The article focuses on the topics: Fuzzy number & Fuzzy set operations.
Citations
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Journal ArticleDOI
TL;DR: Within the effectively formalized representation developed here, based on a complete logical system, it is possible to reconstruct numerous well-known properties of CRI-related fuzzy inference methods, albeit not from the analytic point of view as usually presented, but as formal derivations of the logical system employed.

95 citations

Journal ArticleDOI
TL;DR: This paper shows that the BK-subproduct-based FRI is as effective and efficient as the CRI itself, and suggests a hierarchical inferencing scheme.
Abstract: Fuzzy relational inference (FRI) systems form an important part of approximate reasoning schemes using fuzzy sets. The compositional rule of inference (CRI), which was introduced by Zadeh, has attracted the most attention so far. In this paper, we show that the FRI scheme that is based on the Bandler-Kohout (BK) subproduct, along with a suitable realization of the fuzzy rules, possesses all the important properties that are cited in favor of using CRI, viz., equivalent and reasonable conditions for their solvability, their interpolative properties, and the preservation of the indistinguishability that may be inherent in the input fuzzy sets. Moreover, we show that under certain conditions, the equivalence of first-infer-then-aggregate (FITA) and first-aggregate-then-infer (FATI) inference strategies can be shown for the BK subproduct, much like in the case of CRI. Finally, by addressing the computational complexity that may exist in the BK subproduct, we suggest a hierarchical inferencing scheme. Thus, this paper shows that the BK-subproduct-based FRI is as effective and efficient as the CRI itself.

76 citations


Cites background from "System of fuzzy relation equations ..."

  • ...2) Perfilieva and Lehmke [28] and Perfilieva and Novák [30] have dealt with the continuity of a fuzzy function f ◦R that is adjoint to the CRI mechanism ◦ in a structure S = (X,Y, {Ai,Bi}i=1,...,n ,L, ◦) and a fuzzy relation R that models fuzzy rules (3)....

    [...]

  • ...The main result by Perfilieva and Lehmke [28] and Perfilieva and Novák [30] concerns the relationship of the aforementioned continuity and the interpolativity for the CRI as contained in the following result....

    [...]

  • ...The main result by Perfilieva and Lehmke [28] and Perfilieva and Novák [30] concerns the relationship of the aforementioned continuity and the interpolativity for the CRI as contained in the following result....

    [...]

  • ...2) Perfilieva and Lehmke [28] and Perfilieva and Novák [30] have dealt with the continuity of a fuzzy function f ◦ R that is adjoint to the CRI mechanism ◦ in a structure S = (X,Y, {Ai,Bi}i=1,....

    [...]

  • ...Perfilieva and Lehmke [28] and Perfilieva and Novák [30] have dealt with the continuity of a fuzzy function f ◦ R that is adjoint to the CRI mechanism and a fuzzy relation that models fuzzy rules (3)....

    [...]

Journal ArticleDOI
TL;DR: Novel algorithms are presented for determining minimal solutions for the equations of fuzzy relation equations with continuous t-norms in the form ATR=B, where A and B are the fuzzy subsets of X and Y, respectively.

67 citations

Journal ArticleDOI
TL;DR: Fuzzy inequalities enable us to attain the optimal points that are better solutions than those resulting from the resolution of the similar problems with Ordinary Inequality constraints and an algorithm to generate such optimal solutions is presented.

61 citations

Journal ArticleDOI
TL;DR: It is shown that a solution set comprises both attainable and unattainable solutions, which are closely related to minimal solutions to the equations.

59 citations

References
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Journal ArticleDOI
TL;DR: Fuzzy logic is used to convert heuristic control rules stated by a human operator into an automatic control strategy, and the control strategy set up linguistically proved to be far better than expected in its own right.
Abstract: This paper describes an experiment on the “linguistic” synthesis of a controller for a model industrial plant (a steam engine). Fuzzy logic is used to convert heuristic control rules stated by a human operator into an automatic control strategy. The experiment was initiated to investigate the possibility of human interaction with a learning controller. However, the control strategy set up linguistically proved to be far better than expected in its own right, and the basic experiment of linguistic control synthesis in a non-learning controller is reported here.

6,392 citations

Book
01 Dec 2000

3,353 citations

Book
31 Aug 1998
TL;DR: This paper presents a meta-analysis of many-Valued Propositional Logic, focusing on the part of Lukasiewicz's Logic that deals with Complexity, Undecidability and Generalized Quantifiers and Modalities.
Abstract: Preface. 1. Preliminaries. 2. Many-Valued Propositional Calculi. 3. Lukasiewicz Propositional Logic. 4. Product Logic, Godel Logic. 5. Many-Valued Predicate Logics. 6. Complexity and Undecidability. 7. On Approximate Inference. 8. Generalized Quantifiers and Modalities. 9. Miscellanea. 10. Historical Remarks. References. Index.

3,207 citations

Journal ArticleDOI
TL;DR: This paper provides a methodology for solution of certain basic fuzzy relational equations, with fuzzy sets defined as mappings from sets into complete Brouwerian lattices, covering a large class of types of fuzzy sets.
Abstract: This paper provides a methodology for solution of certain basic fuzzy relational equations, with fuzzy sets defined as mappings from sets into complete Brouwerian lattices, covering a large class of types of fuzzy sets.

986 citations

Book
31 Aug 1999
TL;DR: Fuzzy Logic: What, Why, for Which?
Abstract: Preface. 1. Fuzzy Logic: What, Why, for Which? 2. Algebraic Structures for Logical Calculi. 3. Logical Calculi and Model Theory. 4. Fuzzy Logic in Narrow Sense. 5. Functional Systems in Fuzzy Logic Theories. 6. Fuzzy Logic in Broader Sense. 7. Topoi and Categories of Fuzzy Sets. 8. Few Historical and Concluding Remarks. References. Index.

898 citations