Paper:

# Multi-Level Control of Fuzzy-Constraint Propagation via Evaluations with Linguistic Truth Values in Generalized-Mean-Based Inference

## Kiyohiko Uehara^{*} and Kaoru Hirota^{**}

^{*}Ibaraki University

Hitachi 316-8511, Japan

^{**}Beijing Institute of Technology

Beijing 100081, China

A method is proposed for fuzzy inference which can propagate convex fuzzy-constraints from given facts to consequences in various forms by applying a number of fuzzy rules, particularly when asymmetric fuzzy sets are used for given facts and/or fuzzy rules. The conventionalmethod, α-GEMS (α-level-set and generalized-mean-based inference in synergy with composition), cannot be performed with asymmetric fuzzy sets; it can be conducted only with symmetric fuzzy sets. In order to cope with asymmetric fuzzy sets as well as symmetric ones, a control scheme is proposed for the fuzzy-constraint propagation, which is α-cut based and can be performed independently at each level of α. It suppresses an excessive specificity decrease in consequences, particularly stemming from the asymmetricity. Thereby, the fuzzy constraints of given facts are reflected to those of consequences, to a feasible extent. The theoretical aspects of the control scheme are also presented, wherein the specificity of the support sets of consequences is evaluated via linguistic truth values (LTVs). The proposed method is named α-GEMST (α-level-set and generalized-meanbased inference in synergy with composition via LTV control) in order to differentiate it from α-GEMS. Simulation results show that α-GEMST can be properly performed, particularly with asymmetric fuzzy sets. α-GEMST is expected to be applied to the modeling of given systems with various fuzzy input-output relations.

Full Text (0.3MB)

Would you consider donation?

- [1] L. A. Zadeh, “Fuzzy logic = Computing with words,” IEEE Trans. Fuzzy Syst., Vol.4, No.2, pp. 103-111, 1996.
- [2] L. A. Zadeh, “Inference in fuzzy logic via generalized constraint propagation,” Proc. 1996 26
^{th}Int. Symp. on Multi-Valued Logic (ISMVL’96), pp. 192-195, 1996. - [3] A. Kaufmann, “Introduction to the theory of fuzzy subsets,” New York: Academic, Vol.1, 1975.
- [4] N. R. Pal and J. C. Bezdek, “Measuring Fuzzy Uncertainty,” IEEE Trans. Fuzzy Syst., Vol.2, No.2, pp. 107-118, 1994.
- [5] R. R. Yager, “On the specificity of a possibility distribution,” Fuzzy Sets Syst., Vol.50, pp. 279-292, 1992.
- [6] R. R. Yager, “Measuring tranquility and anxiety in decision making: an application of fuzzy sets,” Int. J. General Systems, Vol.8, pp. 139-146, 1982.
- [7] K. Uehara and M. Fujise, “Multistage fuzzy inference formulated as linguistic-truth-value propagation and its learning algorithm based on back-propagating error information,” IEEE Trans. Fuzzy Syst., Vol.1, No.3, pp. 205-221, Aug. 1993.
- [8] H. Maeda, M. Imuro, and S. Murakami, “A study on the spread of fuzziness in multi-fold multi-stage approximating reasoning – Approximate reasoning with triangular type membership function –,” J. of Japan Society for Fuzzy Theory and Systems, Vol.7, No.1, pp. 113-130, 1995 (in Japanese).
- [9] M. Imuro and H. Maeda, “On the spread of fuzziness in multi-fold and multi-stage fuzzy reasoning,” Proc. 8
^{th}Fuzzy System Symp., Hiroshima, Japan, pp. 221-224, May 1992 (in Japanese). - [10] G. Cheng and Y. Fu, “Error Estimation of Perturbation under CRI,” newblock IEEE Trans. Fuzzy Syst., newblock Vol.14, No.6, pp. 709-715, Dec. 2006.
- [11] S. Aja-Fernández and C. Alberola-L’opez, “Fast inference using transition matrices: An extension to nonlinear operators,” newblock IEEE Trans. Fuzzy Syst., newblock Vol.13, No.4, pp. 478-490, Aug. 2005.
- [12] S. J. Chen and S. M. Chen, “Fuzzy risk analysis based on similarity measures of generalized fuzzy numbers,” newblock IEEE Trans. Fuzzy Syst., newblock Vol.11, No.1, pp. 45-56, Feb. 2003.
- [13] J. Casillas, O. Cord’on, M. J. del Jesus, and F. Herrera, “Genetic tuning of fuzzy rule deep structures preserving interpretability and its interaction with fuzzy rule set reduction,” newblock IEEE Trans. on Fuzzy Systems, Vol.13, No.1, Feb. 2005.
- [14] K. Uehara and K. Hirota, newblock “Parallel fuzzy inference based on α-level sets and generalized means,” newblock Int. J. of Information Sciences, newblock Vol.100, No.1–4, pp. 165-206, Aug. 1997.
- [15] K. Uehara, T. Koyama, and K. Hirota, newblock “Fuzzy Inference with Schemes for Guaranteeing Convexity and Symmetricity in Consequences Based on α-Cuts,” newblock J. of Advanced Computational Intelligence and Intelligent Informatics (JACIII), newblock Vol.13, No.2, pp. 135-149, 2009.
- [16] K. Uehara, T. Koyama, and K. Hirota, “Inference with Governing Schemes for Propagation of Fuzzy Convex Constraints Based on α-Cuts,” J. of Advanced Computational Intelligence and Intelligent Informatics (JACIII), Vol.13, No.3, pp. 321-330, 2009.
- [17] K. Uehara, T. Koyama, and K. Hirota, newblock “Inference Based on α-Cut and Generalized Mean with Fuzzy Tautological Rules,” newblock J. of Advanced Computational Intelligence and Intelligent Informatics (JACIII), newblock Vol.14, No.1, pp. 76-88, 2010.
- [18] K. Uehara, T. Koyama, and K. Hirota, newblock “Suppression Effect of α-Cut Based Inference on Consequence Deviations,” newblock J. of Advanced Computational Intelligence and Intelligent Informatics (JACIII), newblock Vol.14, No.3, pp. 256-271, Apr. 2010.
- [19] K. Uehara and K. Hirota, newblock “Multi-Level Control of Fuzzy-Constraint Propagation in Inference Based on α-Cuts and Generalized Mean,” newblock J. of Advanced Computational Intelligence and Intelligent Informatics (JACIII), newblock Vol.17, No.4, pp. 647-662, July 2013.
- [20] K. Uehara, newblock “Fuzzy Inference Based on a Weighted Average of Fuzzy Sets and its Learning Algorithm for Fuzzy Exemplars,” Proc. of the Int. Joint Conf. of the 4
^{th}IEEE Int. Conf. on Fuzzy Systems and the 2^{nd}Int. Fuzzy Engineering Symp. (FUZZ-IEEE/IFES’95), Vol.IV, pp. 2253-2260, 1995. - [21] newblock S. Wang, T. Tsuchiya, and M. Mizumoto, “Distance-Type Fuzzy Reasoning Method,” J. of Biomedical Fuzzy Systems Association, Vol.1, No.1, pp. 61-78, 1999 (in Japaneses).
- [22] newblock S. Wang, T. Tsuchiya, and M. Mizumoto, “A Learning Algorithm for Distance-Type Fuzzy Reasoning Method,” Biomedical Soft Computing and Human Sciences, Vol.6, No.1, pp. 61-68, 2000.
- [23] K. Uehara, S. Sato, and K. Hirota, newblock “Inference for Nonlinear Mapping with Sparse Fuzzy Rules Based on Multi-Level Interpolation,” newblock J. of Advanced Computational Intelligence and Intelligent Informatics (JACIII), newblock Vol.15, No.3, pp. 264-287, May 2011.
- [24] M. Mizumoto and H.-J. Zimmermann, “Comparison of Fuzzy Reasoning Methods,” Fuzzy Sets Syst., Vol.8, No.3, pp. 253-283, 1982.
- [25] S. Fukami, M. Mizumoto, and K. Tanaka, “Some Considerations on Fuzzy Conditional Inference,” Fuzzy Sets Syst., Vol.4, No.3, pp. 243-273, 1980.
- [26] K. Uehara and M. Fujise, “Fuzzy Inference Based on Families of α-Level Sets,” newblock IEEE Trans. on Fuzzy Systems, Vol.1, No.2, pp. 111-124, May 1993.