Skip to main content

Symmetrical Piecewise Linear Functions Composed by Absolute Value Function Cover

.blurhash-client-img { display: none !important; }

Symmetrical Piecewise Linear Functions Composed by Absolute Value Function

Formalized Mathematics

Volume 31 (2023): Issue 1 (September 2023)

By: Takashi Mitsuishi

Open Access

|Dec 2023

Abstract

We continue the formal development of the application of piecewise linear functions and centroids in the area of fuzzy set theory. The corresponding piecewise linear functions are symmetrical and composed by absolute function. In this paper we prove that the membership functions of isosceles triangle type and isosceles trapezoid type can be constructed by functions of this type.

References

Didier Dubois and Henri Prade. Operations on fuzzy numbers. International Journal of Systems Science, 9(6):613–626, 1978. doi:10.1080/00207727808941724.
Search in Google Scholar Back to article
Didier Dubois and Henri Prade. Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York, 1980.
Search in Google Scholar Back to article
Ronald E. Giachetti and Robert E. Young. A parametric representation of fuzzy numbers and their arithmetic operators. Fuzzy Sets and Systems, 91(2):185–202, 1997. doi:10.1016/S0165-0114(97)00140-1.
Search in Google Scholar Back to article
Eikou Gonda, Hitoshi Miyata, and Masaaki Ohkita. Self-turning of fuzzy rules with di erent types of MSFs (in Japanese). Journal of Japan Society for Fuzzy Theory and Intelligent Informatics, 16(6):540–550, 2004. doi:10.3156/jsoft.16.540.
Search in Google Scholar Back to article
Adam Grabowski. The formal construction of fuzzy numbers. Formalized Mathematics, 22(4):321–327, 2014. doi:10.2478/forma-2014-0032.
Search in Google Scholar Back to article
Adam Grabowski. Fuzzy implications in the Mizar system. In 30th IEEE International Conference on Fuzzy Systems, FUZZ-IEEE 2021, Luxembourg, July 11–14, 2021, pages 1–6. IEEE, 2021. doi:10.1109/FUZZ45933.2021.9494593.
Search in Google Scholar Back to article
Adam Grabowski. On the computer certification of fuzzy numbers. In M. Ganzha, L. Maciaszek, and M. Paprzycki, editors, 2013 Federated Conference on Computer Science and Information Systems (FedCSIS), Federated Conference on Computer Science and Information Systems, pages 51–54, 2013.
Search in Google Scholar Back to article
Adam Grabowski and Takashi Mitsuishi. Initial comparison of formal approaches to fuzzy and rough sets. In Leszek Rutkowski, Marcin Korytkowski, Rafal Scherer, Ryszard Tadeusiewicz, Lotfi A. Zadeh, and Jacek M. Zurada, editors, Artificial Intelligence and Soft Computing – 14th International Conference, ICAISC 2015, Zakopane, Poland, June 14-18, 2015, Proceedings, Part I, volume 9119 of Lecture Notes in Computer Science, pages 160–171. Springer, 2015. doi:10.1007/978-3-319-19324-3 15.
Search in Google Scholar Back to article
Adam Grabowski and Christoph Schwarzweller. On duplication in mathematical repositories. In Serge Autexier, Jacques Calmet, David Delahaye, Patrick D. F. Ion, Laurence Rideau, Renaud Rioboo, and Alan P. Sexton, editors, Intelligent Computer Mathematics, 10th International Conference, AISC 2010, 17th Symposium, Calculemus 2010, and 9th International Conference, MKM 2010, Paris, France, July 5–10, 2010. Proceedings, volume 6167 of Lecture Notes in Computer Science, pages 300–314. Springer, 2010. doi:10.1007/978-3-642-14128-7 26.
Search in Google Scholar Back to article
Adam Grabowski and Christoph Schwarzweller. Translating mathematical vernacular into knowledge repositories. In Michael Kohlhase, editor, Mathematical Knowledge Management, volume 3863 of Lecture Notes in Computer Science, pages 49–64. Springer, 2006. doi:10.1007/11618027 4. 4th International Conference on Mathematical Knowledge Management, Bremen, Germany, MKM 2005, July 15–17, 2005, Revised Selected Papers.
Search in Google Scholar Back to article
Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Mizar in a nutshell. Journal of Formalized Reasoning, 3(2):153–245, 2010.
Search in Google Scholar Back to article
Tetsuro Katafuchi, Kiyoji Asai, and Hiroshi Fujita. Investigation of de uzification in fuzzy inference: Proposal of a new defuzzification method (in Japanese). Medical Imaging and Information Sciences, 18(1):19–30, 2001. doi:10.11318/mii1984.18.19.
Search in Google Scholar Back to article
Ebrahim H. Mamdani. Application of fuzzy algorithms for control of simple dynamic plant. IEE Proceedings, 121:1585–1588, 1974.
Search in Google Scholar Back to article
Takashi Mitsuishi. Some properties of membership functions composed of triangle functions and piecewise linear functions. Formalized Mathematics, 29(2):103–115, 2021. doi:10.2478/forma-2021-0011.
Search in Google Scholar Back to article
Takashi Mitsuishi. Definition of centroid method as defuzzification. Formalized Mathematics, 30(2):125–134, 2022. doi:10.2478/forma-2022-0010.
Search in Google Scholar Back to article
Takashi Mitsuishi. Isosceles triangular and isosceles trapezoidal membership functions using centroid method. Formalized Mathematics, 31(1):59–66, 2023. doi:10.2478/forma-2023-0006.
Search in Google Scholar Back to article
Takashi Mitsuishi, Takanori Terashima, Nami Shimada, Toshimichi Homma, and Yasunari Shidama. Approximate reasoning using LR fuzzy number as input for sensorless fuzzy control. In 2016 IEEE Symposium on Sensorless Control for Electrical Drives (SLED), pages 1–5, 2016. doi:10.1109/SLED.2016.7518804.
Search in Google Scholar Back to article
Masaharu Mizumoto. Improvement of fuzzy control (IV)-case by product-sum-gravity method. In Proc. 6th Fuzzy System Symposium, 1990, pages 9–13, 1990.
Search in Google Scholar Back to article
Timothy J. Ross. Fuzzy Logic with Engineering Applications. John Wiley and Sons Ltd, 2010.
Search in Google Scholar Back to article
Luciano Stefanini and Laerte Sorini. Fuzzy arithmetic with parametric LR fuzzy numbers. In Proceedings of the Joint 2009 International Fuzzy Systems Association World Congress and 2009 European Society of Fuzzy Logic and Technology Conference, pages 600–605, 2009.
Search in Google Scholar Back to article
Werner Van Leekwijck and Etienne E. Kerre. Defuzzification: Criteria and classification. Fuzzy Sets and Systems, 108(2):159–178, 1999.
Search in Google Scholar Back to article
Lotfi Zadeh. Fuzzy sets. Information and Control, 8(3):338–353, 1965. doi:10.1016/S0019-9958(65)90241-X.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/forma-2023-0024 | Journal eISSN: 1898-9934 | Journal ISSN: 1426-2630

Journal RSS Feed

Language: English

Page range: 299 - 308

Accepted on: Dec 18, 2023

|

Published on: Dec 31, 2023

Published by: University of Białystok

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

Related subjects:

Logic and set theory,

Computer sciences,

Artificial intelligence

© 2023 Takashi Mitsuishi, published by University of Białystok
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 31 (2023): Issue 1 (September 2023)

Previous article