On a functional equation connected to the distributivity of fuzzy implications over triangular norms and conorms

Michał Baczyński; Tomasz Szostok; Wanda Niemyska

Kybernetika (2014)

  • Volume: 50, Issue: 5, page 679-695
  • ISSN: 0023-5954

Abstract

top
Distributivity of fuzzy implications over different fuzzy logic connectives have a very important role to play in efficient inferencing in approximate reasoning, especially in fuzzy control systems (see [9, 15] and [4]). Recently in some considerations connected with these distributivity laws, the following functional equation appeared (see [5]) f ( min ( x + y , a ) ) = min ( f ( x ) + f ( y ) , b ) , where a , b > 0 and f : [ 0 , a ] → [ 0 , b ] is an unknown function. In this paper we consider in detail a generalized version of this equation, namely the equation f ( m 1 ( x + y ) ) = m 2 ( f ( x ) + f ( y ) ) , where m 1 , m 2 are functions defined on some intervals of ℝ satisfying additional assumptions. We analyze the cases when m 2 is injective and when m 2 is not injective.

How to cite

top

Baczyński, Michał, Szostok, Tomasz, and Niemyska, Wanda. "On a functional equation connected to the distributivity of fuzzy implications over triangular norms and conorms." Kybernetika 50.5 (2014): 679-695. <http://eudml.org/doc/262200>.

@article{Baczyński2014,
abstract = {Distributivity of fuzzy implications over different fuzzy logic connectives have a very important role to play in efficient inferencing in approximate reasoning, especially in fuzzy control systems (see [9, 15] and [4]). Recently in some considerations connected with these distributivity laws, the following functional equation appeared (see [5]) \[ f(\min (x+y,a))=\min (f(x)+f(y),b), \] where $a,b>0$ and $f\colon [0,a]\rightarrow [0,b]$ is an unknown function. In this paper we consider in detail a generalized version of this equation, namely the equation \[ f(m\_1(x+y))=m\_2(f(x)+f(y)), \] where $m_1,m_2$ are functions defined on some intervals of $\{\mathbb \{R\}\}$ satisfying additional assumptions. We analyze the cases when $m_2$ is injective and when $m_2$ is not injective.},
author = {Baczyński, Michał, Szostok, Tomasz, Niemyska, Wanda},
journal = {Kybernetika},
keywords = {fuzzy connectives; fuzzy implication; distributivity; functional equations; fuzzy connectives; fuzzy implication; distributivity; functional equations},
language = {eng},
number = {5},
pages = {679-695},
publisher = {Institute of Information Theory and Automation AS CR},
title = {On a functional equation connected to the distributivity of fuzzy implications over triangular norms and conorms},
url = {http://eudml.org/doc/262200},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Baczyński, Michał
AU - Szostok, Tomasz
AU - Niemyska, Wanda
TI - On a functional equation connected to the distributivity of fuzzy implications over triangular norms and conorms
JO - Kybernetika
PY - 2014
PB - Institute of Information Theory and Automation AS CR
VL - 50
IS - 5
SP - 679
EP - 695
AB - Distributivity of fuzzy implications over different fuzzy logic connectives have a very important role to play in efficient inferencing in approximate reasoning, especially in fuzzy control systems (see [9, 15] and [4]). Recently in some considerations connected with these distributivity laws, the following functional equation appeared (see [5]) \[ f(\min (x+y,a))=\min (f(x)+f(y),b), \] where $a,b>0$ and $f\colon [0,a]\rightarrow [0,b]$ is an unknown function. In this paper we consider in detail a generalized version of this equation, namely the equation \[ f(m_1(x+y))=m_2(f(x)+f(y)), \] where $m_1,m_2$ are functions defined on some intervals of ${\mathbb {R}}$ satisfying additional assumptions. We analyze the cases when $m_2$ is injective and when $m_2$ is not injective.
LA - eng
KW - fuzzy connectives; fuzzy implication; distributivity; functional equations; fuzzy connectives; fuzzy implication; distributivity; functional equations
UR - http://eudml.org/doc/262200
ER -

References

top
  1. Baczyński, M., 10.1142/S0218488501000764, Int. J. Uncertain. Fuzziness Knowledge-Based Systems 9 (2001), 229-238. Zbl1113.03315MR1821991DOI10.1142/S0218488501000764
  2. Baczyński, M., On the distributivity of fuzzy implications over continuous and Archimedean triangular conorms., Fuzzy Sets and Systems 161 (2010), 1406-1419. Zbl1204.03029MR2606422
  3. Baczyński, M., On the distributivity of fuzzy implications over representable uninorms., Fuzzy Sets and Systems 161 (2010), 2256-2275. Zbl1252.03046MR2658032
  4. Baczyński, M., Jayaram, B., Fuzzy Implications., Studies in Fuzziness and Soft Computing 231, Springer, Berlin Heidelberg 2008. Zbl1293.03012
  5. Baczyński, M., Jayaram, B., 10.1109/TFUZZ.2008.924201, IEEE Trans. Fuzzy Syst. 17 (2009), 590-603. DOI10.1109/TFUZZ.2008.924201
  6. Baczyński, M., Qin, F., 10.1016/j.ijar.2012.10.001, Int. J. Approx. Reason. 54 (2013), 290-296. Zbl1280.03029MR3021572DOI10.1016/j.ijar.2012.10.001
  7. Baczyński, M., Szostok, T., Niemyska, W., On a functional equation related to distributivity of fuzzy implications., In: 2013 IEEE International Conference on Fuzzy Systems (FUZZ IEEE 2013) Hyderabad 2013, pp. 1-5. 
  8. Balasubramaniam, J., Rao, C. J. M., 10.1109/TFUZZ.2004.825075, IEEE Trans. Fuzzy Syst. 12 (2004), 194-198. DOI10.1109/TFUZZ.2004.825075
  9. Combs, W. E., Andrews, J. E., 10.1109/91.660804, IEEE Trans. Fuzzy Syst. 6 (1998), 1-11. DOI10.1109/91.660804
  10. Combs, W. E., 10.1109/TFUZZ.1999.771094, IEEE Trans. Fuzzy Syst. 7 (1999), 371-373. DOI10.1109/TFUZZ.1999.771094
  11. Combs, W. E., 10.1109/TFUZZ.1999.784215, IEEE Trans. Fuzzy Syst. 7 (1999), 477-478. DOI10.1109/TFUZZ.1999.784215
  12. Baets, B. De, Fuzzy morphology: A logical approach., In: Uncertainty Analysis in Engineering and Science: Fuzzy Logic, Statistics, and Neural Network Approach (B. M. Ayyub and M. M. Gupta, eds.), Kluwer Academic Publishers, Norwell 1997, pp. 53-68. Zbl1053.03516
  13. Dick, S., Kandel, A., 10.1109/91.784213, IEEE Trans. Fuzzy Syst. 7 (1999), 475-477. DOI10.1109/91.784213
  14. González-Hidalgo, M., Massanet, S., Mir, A., Ruiz-Aguilera, D., Fuzzy hit-or-miss transform using the fuzzy mathematical morphology based on T-norms., In: Aggregation Functions in Theory and in Practise (H. Bustince et al., eds.), Advances in Intelligent Systems and Computing 228, Springer, Berlin - Heidelberg 2013, pp. 391-403. Zbl1277.68283
  15. Jayaram, B., 10.1016/j.ijar.2007.07.009, Int. J. Approx. Reason. 48 (2008), 156-173. Zbl1184.68511MR2420665DOI10.1016/j.ijar.2007.07.009
  16. Klement, E. P., Mesiar, R., Pap, E., Triangular Norms., Kluwer Academic Publishers, Dordrecht 2000. Zbl1087.20041MR1790096
  17. Kuczma, M., An Introduction to the Theory of Functional Equations and Inequalities. Cauchy's Equation and Jensen's Inequality., Państwowe Wydawnictwo Naukowe (Polish Scientific Publishers) and Uniwersytet Śląski, Warszawa-Kraków-Katowice 1985. Zbl1221.39041MR0788497
  18. Ling, C. H., Representation of associative functions., Publ. Math. Debrecen 12 (1965), 189-212. Zbl0137.26401MR0190575
  19. Mendel, J. M., Liang, Q., 10.1109/91.771093, IEEE Trans. Fuzzy Syst. 7 (1999), 369-371. DOI10.1109/91.771093
  20. Qin, F., Baczyński, M., Xie, A., 10.1109/TFUZZ.2011.2171188, IEEE Trans. Fuzzy Syst. 20 (2012), 153-167. DOI10.1109/TFUZZ.2011.2171188
  21. Qin, F., Yang, L., 10.1016/j.ijar.2010.07.005, Int. J. Approx. Reason. 51 (2010), 984-992. Zbl1226.03036MR2719614DOI10.1016/j.ijar.2010.07.005
  22. Ruiz-Aguilera, D., Torrens, J., Distributivity of strong implications over conjunctive and disjunctive uninorms., Kybernetika 42 (2006), 319-336. Zbl1249.03030MR2253392
  23. Ruiz-Aguilera, D., Torrens, J., Distributivity of residual implications over conjunctive and disjunctive uninorms., Fuzzy Sets and Systems 158 (2007), 23-37. Zbl1114.03022MR2287424
  24. Trillas, E., Alsina, C., On the law [ ( p ∧ q ) → r ] = [ ( p → r ) ∨ ( q → r ) ] in fuzzy logic., IEEE Trans. Fuzzy Syst. 10 (2002), 84-88. 

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.