Paper:

# Multi-Level Control of Fuzzy-Constraint Propagation in Inference with Fuzzy Rule Interpolation at an Infinite Number of Activating Points

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

^{*}Ibaraki University

Hitachi 316-8511, Japan

^{**}Beijing Institute of Technology

Beijing 100081, China

An inference method is proposed, which can control the degree to which the fuzzy constraints of given facts are propagated to those of consequences via the nonlinear mapping represented by fuzzy rules. The conventional method, α-GEMST (α-level-set and generalized-mean-based inference in synergy with composition via linguistic-truth-value control), has limitations in the control of the propagation degree. In contrast, the proposed method can fully control the fuzzy-constraint propagation to a different degree with each fuzzy rule. After the nonlinear mapping, the proposed method performs fuzzy-logic-based control for further fuzzy-constraint propagation wherein evaluations are conducted via linguistic truth values to suppress the excessive specificity decrease in deduced consequences. Thereby, fuzzy constraints can be propagated in various ways by selecting one pair from the widely available implications and compositional operations. The proposed method controls the fuzzy-constraint propagation at the multi-levels of α in its α-cut-based operations. This scheme contributes to fast computation with parallel processing for each level of α. Simulation results illustrate that the proposed method can properly control the propagation degree. The proposed method is expected to be applied to the modeling of given systems with various fuzzy input-output relations.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.21, No.3, pp. 425-447, 2017.

- [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 26th 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] I. B. Turksen and Y. Tian, “Combination of rules or their consequences in fuzzy expert systems,” Fuzzy Sets Syst., Vol.58, No.1, pp. 3-40, 1993.
- [8] G. Cheng and Y. Fu, “Error estimation of perturbation under CRI,” IEEE Trans. Fuzzy Syst., Vol.14, No.6, pp. 709-715, Dec. 2006.
- [9] 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.
- [10] 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).
- [11] M. Imuro and H. Maeda, “On the spread of fuzziness in multi-fold and multi-stage fuzzy reasoning,” Proc. 8th Fuzzy System Symp., Hiroshima, Japan, May 1992, pp. 221-224 (in Japanese).
- [12] S. Aja-Fernández and C. Alberola-López, “Fast inference using transition matrices: An extension to nonlinear operators,” IEEE Trans. Fuzzy Syst., Vol.13, No.4, pp. 478-490, Aug. 2005.
- [13] S. J. Chen and S. M. Chen, “Fuzzy risk analysis based on similarity measures of generalized fuzzy numbers,” IEEE Trans. Fuzzy Syst., Vol.11, No.1, pp. 45-56, Feb. 2003.
- [14] J. Casillas, O. Cordón, M. J. del Jesus, and F. Herrera, “Genetic tuning of fuzzy rule deep structures preserving interpretability and its interaction with fuzzy rule set reduction,” IEEE Trans. Fuzzy Syst., Vol.13, No.1, Feb. 2005.
- [15] K. Uehara and K. Hirota, “Parallel fuzzy inference based on α-level sets and generalized means,” Int. J. of Information Sciences, Vol.100, No.1-4, pp. 165-206, Aug. 1997.
- [16] K. Uehara, T. Koyama, and K. Hirota, “Fuzzy inference with schemes for guaranteeing convexity and symmetricity in consequences based on α-cuts,” J. Adv. Comput. Intell. Intell. Inform. (JACIII), Vol.13, No.2, pp. 135-149, 2009.
- [17] K. Uehara, T. Koyama, and K. Hirota, “Inference with governing schemes for propagation of fuzzy convex constraints based on α-cuts,” J. Adv. Comput. Intell. Intell. Inform. (JACIII), Vol.13, No.3, pp. 321-330, 2009.
- [18] K. Uehara, T. Koyama, and K. Hirota, “Inference based on α-cut and generalized mean with fuzzy tautological rules,” J. Adv. Comput. Intell. Intell. Inform. (JACIII), Vol.14, No.1, pp. 76-88, 2010.
- [19] K. Uehara, T. Koyama, and K. Hirota, “Suppression effect of α-cut based inference on consequence deviations,” J. Adv. Comput. Intell. Intell. Inform. (JACIII), Vol.14, No.3, pp. 256-271, Apr. 2010.
- [20] K. Uehara and K. Hirota, “Multi-level control of fuzzy-constraint propagation via evaluations with linguistic truth values in generalized-mean-based inference,” J. Adv. Comput. Intell. Intell. Inform. (JACIII), Vol.20, No.2, pp. 355-377, 2016.
- [21] K. Uehara and K. Hirota, “Multi-level control of fuzzy-constraint propagation in inference based on α-cuts and generalized mean,” J. Adv. Comput. Intell. Intell. Inform. (JACIII), Vol.17, No.4, pp. 647-662, July 2013.
- [22] K. Uehara and K. Hirota, “Inference with fuzzy rule interpolation at an infinite number of activating points,” J. Adv. Comput. Intell. Intell. Inform. (JACIII), Vol.19, No.1, pp. 74-90, 2015.
- [23] K. Uehara, S. Sato, and K. Hirota, “Inference for nonlinear mapping with sparse fuzzy rules based on multi-level interpolation,” J. Adv. Comput. Intell. Intell. Inform. (JACIII), Vol.15, No.3, pp. 264-287, May 2011.
- [24] K. Uehara and K. Hirota, “Infinite-level interpolation for inference with sparse fuzzy rules: Fundamental analysis toward practical use,” J. Adv. Comput. Intell. Intell. Inform. (JACIII), Vol.17, No.1, pp. 44-59, Jan. 2013.
- [25] L. T. Kóczy and K. Hirota, “Approximate reasoning by linear rule interpolation and general approximation,” Int. J. Approx. Reason., Vol.9, pp. 197-225, 1993.
- [26] L. T. Kóczy and K. Hirota, “Size reduction by interpolation in fuzzy rule bases,” IEEE Trans. Syst., Man, Cybern. B, Cybern., Vol.27, No.1, pp. 14-33, Feb. 1997.
- [27] D. Tikk and P. Baranyi, “Comprehensive analysis of a new fuzzy rule interpolation method,” IEEE Trans. Fuzzy Syst., Vol.8, No.3, pp. 281-296, June 2000.
- [28] P. Baranyi, L. T. Kóczy, and T. D. Gedeon, “A generalized concept for fuzzy rule interpolation,” IEEE Trans. Fuzzy Syst., Vol.12, No.6, pp. 820-837, Dec. 2004.
- [29] K. W. Wong, D. Tikk, T. D. Gedeon, and L. T. Kóczy, “Fuzzy rule interpolation for multidimensional input spaces with applications: A case study,” IEEE Trans. Fuzzy Syst., Vol.13, No.6, pp. 809-819, Dec. 2005.
- [30] Z. Huang and Q. Shen, “Fuzzy interpolative reasoning via scale and move transformation,” IEEE Trans. Fuzzy Syst., Vol.14, No.2, pp. 340-359, April 2006.
- [31] Z. Huang and Q. Shen, “Fuzzy interpolation and extrapolation: A practical approach,” IEEE Trans. Fuzzy Syst., Vol.16, No.1, pp. 13-28, April 2008.
- [32] L. T. Kóczy and Sz. Kovács, “On the preservation of convexity and piecewise linearity in linear fuzzy rule interpolation,” Technical Report, LIFE Chair of Fuzzy Theory, DSS, Tokyo Institute of Technology, Japan, p. 23, 1993.
- [33] L. T. Kóczy and Sz. Kovács, “Shape of the fuzzy conclusion generated by linear interpolation of trapezoidal if ... then rules,” Fuzzy Set Theory and its Applications, Tatra Mountains Mathematical Publications, Mathematical Institute Slovak Academy of Science, Vol.6, pp. 83-93, Bratislava, Slovakia, 1995.
- [34] D. Tikk, Z. C. Johanyák, S. Kovács, and K. W. Wong, “Fuzzy rule interpolation and extrapolation techniques: Criteria and evaluation guidelines,” J. Adv. Comput. Intell. Intell. Inform. (JACIII), Vol.15, No.3, pp. 254-263, 2011.
- [35] Q. Shen, and L. Yang, “Generalization of scale and move transformation-based fuzzy interpolation,” J. Adv. Comput. Intell. Intell. Inform. (JACIII), Vol.15, No.3, pp. 288-298, 2011.
- [36] L. Kovács, “Compound distance function for similarity measurement between fuzzy sets,” J. Adv. Comput. Intell. Intell. Inform. (JACIII), Vol.15, No.3, pp. 299-303, 2011.
- [37] S. Kato and K. W. Wong, “Intelligent automated guided vehicle controller with reverse strategy,” J. Adv. Comput. Intell. Intell. Inform. (JACIII), Vol.15, No.3, pp. 304-312, 2011.
- [38] D. Vincze and S. Kovács, “Performance optimization of the fuzzy rule interpolation method “FIVE”,” J. Adv. Comput. Intell. Intell. Inform. (JACIII), Vol.15, No.3, pp. 313-320, 2011.
- [39] K. Uehara and K. Hirota, “Multi-level interpolation for inference with sparse fuzzy rules: An extended way of generating multi-level points,” J. Adv. Comput. Intell. Intell. Inform. (JACIII), Vol.17, No.2, pp. 127-148, Mar. 2013.
- [40] M. Mizumoto and H.-J. Zimmermann, “Comparison of fuzzy reasoning methods,” Fuzzy Sets Syst., Vol.8, No.3, pp. 253-283, 1982.
- [41] S. Fukami, M. Mizumoto, and K. Tanaka, “Some considerations on fuzzy conditional inference,” Fuzzy Sets Syst., Vol.4, No.3, pp. 243-273, 1980.
- [42] K. Uehara and M. Fujise, “Fuzzy inference based on families of α-level sets,” IEEE Trans. Fuzzy Syst., Vol.1, No.2, pp. 111-124, May 1993.

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