Abstract
Defining the standard Boolean operations on fuzzy Booleans with the compositional rule of inference (CRI) or Zadeh’s extension principle gives counter-intuitive results. We introduce and motivate a slight adaptation of the CRI, which only effects the results for non-normal fuzzy sets. It is shown that the adapted CRI gives the expected results for the standard Boolean operations on fuzzy Booleans. As a second application, we show that the adapted CRI enables a don’t-care value in approximate reasoning. From the close connection between the CRI and Zadeh’s extension principle, we derive an adaptation of the extension principle, which, like the modified CRI, also gives the expected Boolean operations on fuzzy Booleans.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
van den Broek, P.M., Noppen, J.A.R.: Approximate Reasoning with Fuzzy Booleans. In: IPMU 2004, 10th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, Editrice Universita La Sapienza, Italy, pp. 1429–1434 (2004)
Dubois, D., Prade, H.: Fuzzy numbers: an overview. In: Bezdek, J.C. (ed.) Analysis of Fuzzy Information - vol. 1: Mathematics and Logic, pp. 3–39. CRC Press, Boca Raton (1987)
Klir, G.J., Yuan, B.: Fuzzy sets and fuzzy logic, theory and applications. Prentice-Hall, Englewood Cliffs (1995)
Mamdani, E.H., Assilian, S.: An experiment in linguistic synthesis with a fuzzy logic controller. Intern. J. of Man-Machine Studies 7, 1–13 (1975)
Yager, R.R.: A characterization of the extension principle. Fuzzy Sets and Systems 18, 205–217 (1986)
Zadeh, L.: The concept of a linguistic variable and its application to approximate reasoning. Information Sciences 8, 199–251 (1975)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
van den Broek, P., Noppen, J. (2007). The Compositional Rule of Inference and Zadeh’s Extension Principle for Non-normal Fuzzy Sets. In: Castillo, O., Melin, P., Ross, O.M., Sepúlveda Cruz, R., Pedrycz, W., Kacprzyk, J. (eds) Theoretical Advances and Applications of Fuzzy Logic and Soft Computing. Advances in Soft Computing, vol 42. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72434-6_63
Download citation
DOI: https://doi.org/10.1007/978-3-540-72434-6_63
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-72433-9
Online ISBN: 978-3-540-72434-6
eBook Packages: EngineeringEngineering (R0)