Abstract
A fuzzy IF-THEN rule base is considered as a system of equations for a functional operator. We study several conditions so that such a system of functional equations has at most one solution. Several versions of compactness inspired by the compactness of consequence operators used in the theory of formal systems from mathematical logic play a fundamental role in formulating these conditions and in proving corresponding theorems.
This research was supported by the Deutsche Forschungsgemeinschaft as part of the Collabo-rative Research Center ’Computational Intelligence’ (SFB 531)
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J.F. Baldwin and B.W. Pilsworth. Axiomatic approach to implication for approximate reasoning with fuzzy logic. Fuzzy Sets and Systems 3, 1980, pages 193–219.
M. Diehl. Untersuchungen zum Kalkül der Fuzzy-IF-THEN-Regeln. Diplomarbeit, Universität Dortmund, Fachbereich Informatik, Lehrstuhl I, Juli 1994.
S. Fukami, M. Mizumoto, and K. Tanaka. Some considerations of fuzzy conditional inference. Fuzzy Sets and Systems 4, 1980, pages 243–273.
S. Gottwald. On the existence of solutions of systems of fuzzy equations. Fuzzy Sets and Systems 12, 1984, pages 301–302.
S. Gottwald. Characterizations of the solvability of fuzzy equations. Elektron. Informationsverarb. Kybernet. 22, 1986, pages 67–91.
S. Gottwald. On some theoretical results concerning the construction of fuzzy controllers. In St.F. Bocklisch, et all (editors). Fuzzy Sets Applications, Methodological Approaches, and Results. Math. Research, volume 30, Akademie-Verlag, Berlin, 1986, pages 45–55.
S. Gottwald and W. Pedrycz. Solvability of fuzzy relational equations and manipulation of fuzzy data. Fuzzy Sets and Systems 18, 1986, pages 1–21.
S. Gottwald and W. Pedrycz. On the methodology of solving fuzzy relational equations and its impact on fuzzy modelling. In M.M. Gupta, and T. Yamakawa (editors). Fuzzy Logic in Knowledge-Based Systems, Decision and Control. North Holland Publ. Corp., Amsterdam, 1988, pages 197–210.
F. Klawonn and R. Kruse. Equality relations as a basis for fuzzy control. Fuzzy Sets and Systems 54, 1993, pages 147–156.
F. Klawonn and R. Kruse. Fuzzy control as interpolation on the basis of equality relations. Proceedings of the Second IEEE International Conference on Fuzzy Systems, San Francisco, California, March 28–April 1, 1993.
L.T. Kóczy. Approximate reasoning and control with sparse and/or inconsistent fuzzy rule bases. In B. Reusch (editor). Fuzzy Logic — Theorie und Praxis. 2. Dortmunder Fuzzy-Tage, Dortmund, June 9–10, 1992, Springer-Verlag, 1993, pages 42–65.
L.T. Kóczy. Compression of fuzzy rule bases by interpolation. First Asian Fuzzy Systems Symposium, Singapore, November 23–26, 1993. Institute of Systems Science, National Uni-versity of Singapore. Proceedings.
L.T. Kóczy. On the calculus of fuzzy rules. Fifth International Fuzzy Systems Association World Congress’93, Proceedings, volume 1, Seoul, Korea, July 4–9, 1993, pages 1–2.
L.T. Kóczy (editor). Fuzzy Logic, Texts. Volume 2, Fuzzy Systems II (Fuzzy Reasoning and Control), TEMPUS_JEP 07759-94-MODIFY, Budapest 1997.
R. Kruse, J. Gebhardt, and F. Klawonn. Foundations of Fuzzy Systems. Wiley, Chichester, 1994.
S. Lehmke, B. Reusch, K.-H. Temme, and H. Thiele. On interpreting fuzzy IF-THEN rule bases by concepts of functional analysis. Technical Report CI-19/98, University of Dortmund, Collaborative Research Center 531, February 1998.
H. Nakanishi, I.B. Türkşen, and M. Sugeno. A review and comparison of six reasoning methods. Fuzzy Sets and Systems 57, 1993, pages 257–294.
Bernd Reusch, and Helmut Thiele. On semantic interpretations of fuzzy IF-THEN rule bases by concepts of Functional Analysis. Paper. To appear in Fuzzy Sets and Systems.
K.-H. Temme, and H. Thiele. On the correctness of the principles FATI and FITA and their equivalence. IFSA’95 — Sixth International Fuzzy Systems Association World Congress, volume 2, São Paulo, Brazil, July 22–28, 1995, pages 475–478.
H. Thiele. On the calculus of fuzzy IF-THEN rules. In H.-J. Zimmermann, M.G. Negoiţâa, and D. Dascâlu (editors). Real World Applications of Intelligent Technologies. Mangalia, Romania, September 1995, Black Sea University, Editura Academiei Române, pages 91–111.
H. Thiele. On similarity based fuzzy clusterings. 17th Linz Seminar on Fuzzy Set Theory, Johannes Kepler Universität, Linz, Austria, February 20–24, 1996. Abstracts, pages 135–138.
H. Thiele. A characterization of RUSPINI-partitions by similarity relations. IPMU’96, International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, Universidad de Granada, Spain, July 1–5, 1996. Conference Proceedings, vol. I, pages 389–394.
H. Thiele. Zur funktionalanalytischen Begründung des approximativen Schließens und von Fuzzy Control. 16. Workshop “Interdiziplinäre Methoden in der Informatik”, Meinerzhagen-Valbert, 16.–19. September, 1996. Universität Dortmund, Fachbereich Informatik, Lehrstuhl I, Forschungsbericht Nr. 639 des Fachbereichs Informatik, Seiten 1-20.
H. Thiele. Investigating approximate reasoning and fuzzy control by concepts of functional analysis. First International Conference on Conventional and Knowledge-Based Intelligent Electronic Systems, volume 2, Adelaide, Australia, May 21–23, 1997, pages 493–500.
H. Thiele. On closure operators in fuzzy deductive systems and fuzzy algebras. 28th International Symposium on Multiple-Valued Logic, Fukuoka, Japan, May 27–29, 1998. Proceedings, pages 304–309. Extended Version: On Closure Operators in Fuzzy Algebras and Fuzzy Deductive Systems. Technical Report CI-34/98, University of Dortmund, Collaborative Research Center 531 (Computational Intelligence), April 1998, 23 pages.
H. Thiele. Investigating the logical structure of fuzzy IF-THEN rule bases using concepts of mathematical logic and of functional analysis. In M. Jamshidi, et all (editors). World Automation Congress 1998 — Advances in Robotics, Manufacturing, Automation, Control, Soft Computing, Multimedia, and Image Processing (WAC“98): Proceedings. Anchorage, Alaska, USA, May 10–14, 1998, NASA, ISSCI, pages 132.1–132.6.
H. Thiele. Funktionalanalytische Methoden zur Interpretation von Fuzzy Control Regelbasen (Methods of Functional Analysis for Interpreting Fuzzy Control Rule Bases. 9. Workshop “Fuzzy Control” des GMD-Fachausschusses 5.22, Dortmund, Germany, 4.–5. November 1999, Forschungsbericht Nr. 0499 der Fakultät Elektrotechnik, Seiten 51–75.
H. Thiele. On fuzzy clusterings with variable boundary conditions. Fith International Conference on Fuzzy Sets Theory and its Applications (FSTA 2000), Liptovsky Mikulas, The Slovak Republic, January 30–February 4, 2000. To appear in “TATRA MOUNTAINS MATHEMATICAL PUBLICATIONS”.
E.C.C. Tsang, J.W.T. Lee, and D.S. Yeung. Similarity based fuzzy reasoning methods for fuzzy production rules. IFSA’95 — Sixth International Fuzzy Systems Association World Congress, volume 1, São Paulo, Brazil, July 22–28, 1995, pages 157–160.
I.B. Türkşen, and Y. Tian. Combination of rules or their consequences in fuzzy expert systems. Fuzzy Sets and Systems 58, 1993, pages 3–40.
L.A. Zadeh. Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans. on Systems, Man and Cybernetics 3(1), 1973, pages 28–44.
Z. Zhong, and I.B. Türkşen. An approximate reasoning schema based on similarity measures and interval-valued fuzzy sets. Fuzzy Sets and Systems 34, 1990, pages 323–346.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Thiele, H. (2001). On the Uniqueness of Interpretations for Fuzzy IF-THEN Rule Bases. In: Reusch, B. (eds) Computational Intelligence. Theory and Applications. Fuzzy Days 2001. Lecture Notes in Computer Science, vol 2206. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45493-4_52
Download citation
DOI: https://doi.org/10.1007/3-540-45493-4_52
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42732-2
Online ISBN: 978-3-540-45493-9
eBook Packages: Springer Book Archive