On continuity of the entropy-based differently implicational algorithm

Yiming Tang; Witold Pedrycz

Kybernetika (2019)

  • Volume: 55, Issue: 2, page 307-336
  • ISSN: 0023-5954

Abstract

top
Aiming at the previously-proposed entropy-based differently implicational algorithm of fuzzy inference, this study analyzes its continuity. To begin with, for the FMP (fuzzy modus ponens) and FMT (fuzzy modus tollens) problems, the continuous as well as uniformly continuous properties of the entropy-based differently implicational algorithm are demonstrated for the Tchebyshev and Hamming metrics, in which the R-implications derived from left-continuous t-norms are employed. Furthermore, four numerical fuzzy inference examples are provided, and it is found that the entropy-based differently implicational algorithm can obtain more reasonable solution in contrast with the fuzzy entropy full implication algorithm. Finally, in the entropy-based differently implicational algorithm, we point out that the first fuzzy implication reflects the effect of rule base, and that the second fuzzy implication embodies the inference mechanism.

How to cite

top

Tang, Yiming, and Pedrycz, Witold. "On continuity of the entropy-based differently implicational algorithm." Kybernetika 55.2 (2019): 307-336. <http://eudml.org/doc/294658>.

@article{Tang2019,
abstract = {Aiming at the previously-proposed entropy-based differently implicational algorithm of fuzzy inference, this study analyzes its continuity. To begin with, for the FMP (fuzzy modus ponens) and FMT (fuzzy modus tollens) problems, the continuous as well as uniformly continuous properties of the entropy-based differently implicational algorithm are demonstrated for the Tchebyshev and Hamming metrics, in which the R-implications derived from left-continuous t-norms are employed. Furthermore, four numerical fuzzy inference examples are provided, and it is found that the entropy-based differently implicational algorithm can obtain more reasonable solution in contrast with the fuzzy entropy full implication algorithm. Finally, in the entropy-based differently implicational algorithm, we point out that the first fuzzy implication reflects the effect of rule base, and that the second fuzzy implication embodies the inference mechanism.},
author = {Tang, Yiming, Pedrycz, Witold},
journal = {Kybernetika},
keywords = {fuzzy inference; fuzzy entropy; compositional rule of inference; continuity},
language = {eng},
number = {2},
pages = {307-336},
publisher = {Institute of Information Theory and Automation AS CR},
title = {On continuity of the entropy-based differently implicational algorithm},
url = {http://eudml.org/doc/294658},
volume = {55},
year = {2019},
}

TY - JOUR
AU - Tang, Yiming
AU - Pedrycz, Witold
TI - On continuity of the entropy-based differently implicational algorithm
JO - Kybernetika
PY - 2019
PB - Institute of Information Theory and Automation AS CR
VL - 55
IS - 2
SP - 307
EP - 336
AB - Aiming at the previously-proposed entropy-based differently implicational algorithm of fuzzy inference, this study analyzes its continuity. To begin with, for the FMP (fuzzy modus ponens) and FMT (fuzzy modus tollens) problems, the continuous as well as uniformly continuous properties of the entropy-based differently implicational algorithm are demonstrated for the Tchebyshev and Hamming metrics, in which the R-implications derived from left-continuous t-norms are employed. Furthermore, four numerical fuzzy inference examples are provided, and it is found that the entropy-based differently implicational algorithm can obtain more reasonable solution in contrast with the fuzzy entropy full implication algorithm. Finally, in the entropy-based differently implicational algorithm, we point out that the first fuzzy implication reflects the effect of rule base, and that the second fuzzy implication embodies the inference mechanism.
LA - eng
KW - fuzzy inference; fuzzy entropy; compositional rule of inference; continuity
UR - http://eudml.org/doc/294658
ER -

References

top
  1. Baczyński, M., Jayaram, B., 10.1016/j.fss.2007.11.015, Fuzzy Set Syst. 159 (2008), 1836-1859. MR2428086DOI10.1016/j.fss.2007.11.015
  2. Baczyński, M., Jayaram, B., Fuzzy implications (Studies in Fuzziness and Soft Computing, Vol. 231., Springer, Berlin Heidelberg 2008. MR2428086
  3. Chaudhuria, B. B., Rosenfeldb, A., 10.1016/s0020-0255(99)00037-7, Inform. Sci. 118 (1999), 159-171. MR1723219DOI10.1016/s0020-0255(99)00037-7
  4. Dai, S. S., Pei, D. W., Wang, S. M., 10.1016/j.fss.2011.07.012, Fuzzy Set Syst. 189 (2012), 63-73. MR2871353DOI10.1016/j.fss.2011.07.012
  5. Dai, S. S., Pei, D. W., Guo, D. H., 10.1016/j.ijar.2012.11.007, Int. J. Approx. Reason. 54 (2013), 653-666. MR3041100DOI10.1016/j.ijar.2012.11.007
  6. Liu, F., Zhang, W. G., Fu, J. H., 10.1016/j.ins.2011.09.019, Inform. Sci. 185 (2012), 32-42. DOI10.1016/j.ins.2011.09.019
  7. Liu, F., Zhang, W. G., Wang, Z. X., 10.1016/j.ejor.2011.11.042, Eur. J. Oper. Res. 218 (2012), 747-754. MR2881747DOI10.1016/j.ejor.2011.11.042
  8. Fodor, J. C., 10.1016/0165-0114(94)00210-x, Fuzzy Set Syst. 69 (1995), 141-156. MR1317882DOI10.1016/0165-0114(94)00210-x
  9. Fodor, J., Roubens, M., 10.1007/978-94-017-1648-2, Kluwer Academic Publishers, Dordrecht, 1994. DOI10.1007/978-94-017-1648-2
  10. Gottwald, S., A Treatise on Many-Valued Logics., Research Studies, Studies in Logic and Computation 9, Baldock 2001. Zbl1048.03002MR1856623
  11. Guo, F. F., Chen, T. Y., Xia, Z. Q., Triple I methods for fuzzy reasoning based on maximum fuzzy entropy principle., Fuzzy Syst. Math. 17 (2003), 55-59. MR2026787
  12. Hong, D. H., Hwang, S. Y., 10.1016/0165-0114(94)90107-4, Fuzzy Set Syst. 66 (1994), 383-386. MR1300296DOI10.1016/0165-0114(94)90107-4
  13. Jayaram, B., 10.1109/tfuzz.2007.895969, IEEE Trans. Fuzzy Syst. 16 (2008), 130-144. DOI10.1109/tfuzz.2007.895969
  14. Jaynes, E. T., Where do we stand on maximum entropy?, In: The Maximum Entropy Formalism (R. .D. Levine and M. Tribus, eds.), MIT Press, Cambridge 1978, pp. 15-118. MR0521743
  15. Jaynes, E. T., 10.1109/proc.1982.12425, Proc. IEEE, 70 (1982), 939-952. DOI10.1109/proc.1982.12425
  16. Jenei, S., 10.1016/s0165-0114(97)00198-x, Fuzzy Set Syst. 104 (1999), 333-339. MR1688064DOI10.1016/s0165-0114(97)00198-x
  17. Klement, E. P., Mesiar, R., Pap, E., 10.1007/978-94-015-9540-7, Kluwer Academic Publishers, Dordrecht 2000. Zbl1087.20041MR1790096DOI10.1007/978-94-015-9540-7
  18. Li, H. X., 10.1007/s11432-006-0339-9, Sci. China Ser. F-Inf. Sci. 49 (2006), 339-363. MR2250341DOI10.1007/s11432-006-0339-9
  19. Li, H., Lee, E. S., 10.1016/s0898-1221(03)00147-0, Comput. Math. Appl. 45 (2003), 1683-1693. MR1993238DOI10.1016/s0898-1221(03)00147-0
  20. Luo, M. X., Liu, B., 10.1016/j.jlamp.2016.09.006, J. Log. Algebr. Methods 86 (2017), 298-307. MR3575372DOI10.1016/j.jlamp.2016.09.006
  21. Mas, M., Monserrat, M., Torrens, J., Trillas, E., 10.1109/tfuzz.2007.896304, IEEE Trans. Fuzzy Syst. 15 (2007), 1107-1121. DOI10.1109/tfuzz.2007.896304
  22. Novák, V., Perfilieva, I., Močkoř, J., 10.1007/978-1-4615-5217-8, Kluwer Academic Publishes, Boston, Dordrecht 1999. Zbl0940.03028MR1733839DOI10.1007/978-1-4615-5217-8
  23. Pang, L. M., Tay, K. M., Lim, C. P., 10.1109/tfuzz.2016.2540059, IEEE Trans. Fuzzy Syst. 24 (2016), 1455-1463. DOI10.1109/tfuzz.2016.2540059
  24. Pedrycz, W., Granular Computing: Analysis and Design of Intelligent Systems., CRC Press/Francis and Taylor, Boca Raton 2013. 
  25. Pedrycz, W., 10.1201/9781315216737, Fuzzy Set Syst. 274 (2015), 12-17. MR3355341DOI10.1201/9781315216737
  26. Pedrycz, W., Wang, X. M., 10.1109/tfuzz.2015.2453393, IEEE Trans. Fuzzy Syst. 24 (2016), 489-496. DOI10.1109/tfuzz.2015.2453393
  27. Pei, D. W., 10.1016/s0165-0114(02)00053-2, Fuzzy Set Syst. 131 (2002), 297-302. MR1939842DOI10.1016/s0165-0114(02)00053-2
  28. Pei, D. W., 10.1016/j.ins.2007.09.003, Inform. Sci. 178 (2008), 520-530. MR2363234DOI10.1016/j.ins.2007.09.003
  29. Rosenfeld, A., 10.1016/0167-8655(85)90002-9, Pattern Recogn. Lett. 3 (1985), 229-233. DOI10.1016/0167-8655(85)90002-9
  30. Sarkoci, P., Šabo, M., Information boundedness principle in fuzzy inference process., Kybernetika 38 (2002), 327-338. MR1944313
  31. Szmidt, E., Kacprzyk, J., 10.1016/s0165-0114(98)00402-3, Fuzzy Set Syst. 118 (2001), 467-477. Zbl1045.94007MR1809394DOI10.1016/s0165-0114(98)00402-3
  32. Tang, Y. M., 10.1109/fuzz-ieee.2015.7337944, In: Proc. 2015 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2015), Istanbul, pp. 1-8. MR3324890DOI10.1109/fuzz-ieee.2015.7337944
  33. Tang, Y. M., Liu, X. P., 10.1016/j.camwa.2009.11.016, Comput. Math. Appl. 59 (2010), 1965-1984. MR2595972DOI10.1016/j.camwa.2009.11.016
  34. Tang, Y. M., Ren, F. J., 10.1109/fskd.2013.6816179, Iran. J. Fuzzy Syst. 10 (2013), 1-24. MR3154637DOI10.1109/fskd.2013.6816179
  35. Tang, Y. M., Ren, F. J., 10.3233/IFS-141476, J. Intell. Fuzzy Syst. 28 (2015), 1885-1897. MR3324890DOI10.3233/IFS-141476
  36. Tang, Y. M., Ren, F. J., 10.1142/s0219622014500746, Int. J. Inf. Tech. Decis. 16 (2017), 443-471. DOI10.1142/s0219622014500746
  37. Tang, Y. M., Ren, F. J., Chen, Y. X., 10.1109/jsee.2012.00070, J. Syst. Eng. Electron. 23 (2012), 560-573. DOI10.1109/jsee.2012.00070
  38. Tang, Y. M., Yang, X. Z., 10.1016/j.ijar.2013.04.012, Int. J. Approx. Reason. 54 (2013), 1034-1048. MR3081298DOI10.1016/j.ijar.2013.04.012
  39. Tang, Y. M., Yang, X. Z., Yue, F., 10.1109/fskd.2013.6816179, In: Proc. the 10th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD 2013), pp. 125-129. DOI10.1109/fskd.2013.6816179
  40. Wang, L. X., A Course in Fuzzy Systems and Control., Prentice-Hall, Englewood Cliffs, NJ, 1997. 
  41. Wang, G. J., 10.1016/s0020-0255(98)10103-2, Inform. Sci. 117 (1999), 47-88. MR1705095DOI10.1016/s0020-0255(98)10103-2
  42. Wang, G. J., Fu, L., 10.1016/j.camwa.2004.01.019, Comput. Math. Appl. 49 (2005), 923-932. MR2135223DOI10.1016/j.camwa.2004.01.019
  43. Wang, G. J., Zhou, H. J., Introduction to Mathematical Logic and Resolution Principle., Co-published by Science Press and Alpha International Science Ltd., 2009. 
  44. Yang, X. Y., Yu, F. S., Pedrycz, W., 10.1016/j.ijar.2016.10.010, Int. J. Approx. Reasoning 81 (2017), 1-27. MR3589730DOI10.1016/j.ijar.2016.10.010
  45. Zadeh, L. A., 10.1109/tsmc.1973.5408575, IEEE Trans. Syst. Man Cyber. 3 (1973), 28-44. MR0309582DOI10.1109/tsmc.1973.5408575

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.