Skip to main content
SpringerLink
Log in

Approximate reasoning with aggregation functions satisfying GMP rules

  • Published:
Artificial Intelligence Review Aims and scope Submit manuscript

Abstract

To strengthen the effectiveness of approximate reasoning in fuzzy modus ponens (FMP) and fuzzy modus tollens (FMT) problems, three approximate reasoning methods with aggregation functions are developed and their validity are investigated respectively in this paper. We firstly study some properties of fuzzy implication generated by an aggregation function. And then present an A-compositional rule of inference as an extension of Zadeh’s CRI replacing t-norm by aggregation function. The similarity-based approximate reasoning with aggregation function is further discussed. Moreover, we provide the quintuple implication principle method with aggregation function to solve FMP and FMT problems. Finally, the validity of three approximate reasoning approaches is analyzed respectively using GMP rules in detail.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Baczyński M, Jayaram B (2008) Fuzzy implications. Springer, Berlin

    MATH  Google Scholar 

  • Baldwin J (1979) A new approach to approximate reasoning using a fuzzy logic. Fuzzy Sets Syst 2:309–325

    Article  Google Scholar 

  • Baets de B, Kerre EE (1993) The generalized modus ponens and the triangular fuzzy data model. Fuzzy Sets Syst 59:305–317

    Article  MathSciNet  Google Scholar 

  • Bustince H, Pagola M, Mesiar R, Hüllermeier E, Herrera F (2012) Grouping, overlap, and generalized bientropic functions for fuzzy modeling of pairwise comparisons. IEEE Trans Fuzzy Syst 20:405–415

    Article  Google Scholar 

  • Cappelle B, Kerre EE, Ruan D, Vanmassenhove FR (1991) Characterization of binary operations on the unit interval satisfying the generalized modus ponens inference rule. Math Pannon 2(1):105–121

    MathSciNet  MATH  Google Scholar 

  • Cornelis C, Cock M.D, Kerre E.E(2000) The generalized modus ponens in a fuzzy set theoretical framework, In Fuzzy if-then rules in Computational Intelligence, Kluwer academic publishers, pp 37–59

  • Cornelis C, Kerre E.E(2002) A fuzzy inference methodology based on the fuzzification of set inclusion, In Recent advances in Intelligent Paradigms and Applications, Physica verlag, pp 71–88

  • Demirli K, De Baets B (1999) Basic properties of implicators in a residual framework. Tatra Mt Math Publ 16:31–46

    MathSciNet  MATH  Google Scholar 

  • Dimuro GP, Bedregal B (2015) On residual implications derived from overlap functions. Inf Sci 312:78–88

    Article  MathSciNet  Google Scholar 

  • Dimuro GP, Bedregal B, Santiago RHN (2014) On (G, N)-implications derived from grouping functions. Inf Sci 279:1–17

    Article  MathSciNet  Google Scholar 

  • Feng ZQ, Liu CG (2012) On similarity-based approximate reasoning in interval-valued fuzzy environments. Informatica 36(3):255–262

    MathSciNet  Google Scholar 

  • Fodor JC, Keresztfalvi T (1995) Nonstandard conjunctions and implications in fuzzy logic. Int J Approx Reason 12(2):69–84

    Article  MathSciNet  Google Scholar 

  • Grabisch M, Marichal JL, Mesiar R, Pap E (2009) Aggregation functions. Cambridge University Press, New York

    Book  Google Scholar 

  • Grzegorzewski P (2013) Probabilistic implications. Fuzzy Sets Syst 226:53–66

    Article  MathSciNet  Google Scholar 

  • Jayaram B, Mesiar R (2009) On special fuzzy implications. Fuzzy Sets Syst 160:2063–2085

    Article  MathSciNet  Google Scholar 

  • Klement EP, Mesiar R, Pap E (2000) Triangular norms. Kluwer Academic Publishers, Dordrecht, Boston

    Book  Google Scholar 

  • Kolesárová A, Kerre E.E,(2000) Compositional rule of inference based on triangular norms, In Fuzzy if-then rules in computational intelligence, Kluwer academic publishers, pp 61–80

  • Król A(2011) Dependencies between fuzzy conjunctions and implications, Proceedings of the 7th conference of the European society for fuzzy logic and technology (EUSFLAT-11), pp 230–237

  • Lee CC (1990) Fuzzy logic in control systems: fuzzy logic controller. IEEE Trans Syst Man Cybern 20:404–435

    Article  MathSciNet  Google Scholar 

  • Li DC, Qin SJ (2018) Performance analysis of fuzzy systems based on quintuple implications method. Int J Approx Reason 96:20–35

    Article  MathSciNet  Google Scholar 

  • Li YF, Qin KY, He XX, Meng D (2016) Properties of Raha’s similarity-based approximate reasoning method. Fuzzy Sets Syst 294:48–62

    Article  MathSciNet  Google Scholar 

  • Li YM, Shi ZK, Li ZH (2002) Approximation theory of fuzzy systems based upon genuine many-valued implications: MIMO cases. Fuzzy Sets Syst 130:159–174

    Article  MathSciNet  Google Scholar 

  • Lowen R (1978) On fuzzy complements. Inf Sci 14(2):107–113

    Article  MathSciNet  Google Scholar 

  • Mamdani EH (1977) Application of fuzzy logic to approximate reasoning using linguistic synthesis. IEEE Trans Comput 100(12):1182–1191

    Article  Google Scholar 

  • Magrez P, Smets P (1989) Fuzzy modus ponens: a new model suitable for applications in knowldge-based systems. Int J Intell Syst 4:181–200

    Article  Google Scholar 

  • Mas M, Monserrat M, Ruiz-Aguilera D, Torrens J (2016) RU and (U, N)-implications satisfying modus ponens. Int J Approx Reason 73:123–137

    Article  MathSciNet  Google Scholar 

  • Mas M, Monserrat M, Torrens J, Trillas E (2008) A survey on fuzzy implication functions. IEEE Trans Fuzzy Syst 15(6):1107–1121

    Article  Google Scholar 

  • Mizumoto M (1985) Fuzzy reasoning under new compositional rules of inference. Kybernetes 12:107–117

    Article  MathSciNet  Google Scholar 

  • Ouyang Y (2012) On fuzzy implications determined by aggregation operators. Inf Sci 193:153–162

    Article  MathSciNet  Google Scholar 

  • Pei D (2008) Unified full implication algorithms of fuzzy reasoning. Inf Sci 178:520–530

    Article  MathSciNet  Google Scholar 

  • Pradera A, Beliakov G, Bustince H, De Baets B (2016) A review of the relationships between implication, negation and aggregation functions from the point of view of material implication. Inf Sci 329:357–380

    Article  Google Scholar 

  • Raha S, Pal NR, Ray KS (2002) Similarity-based approximate reasoning: methodology and application. IEEE Trans Syst Man Cybern Part A Syst Hum 32(4):41–547

    Article  Google Scholar 

  • Ruan D, Kerre EE (2010) On the extension of the compositional rule of inference. Int J Intell Syst 8(7):807–817

    Article  Google Scholar 

  • Trillas E, Alsina C, Pradera A (2004) On MPT-implication functions for fuzzy logic. Rev R Acad Cien Ser A Mat 98(1):259–271

    MathSciNet  MATH  Google Scholar 

  • Turksen IB, Zhong Z (1988) An approximate analogical reasoning approach based on similarity measures. IEEE Trans Syst Man Cybern 18:1049–1056

    Article  Google Scholar 

  • Wang GJ (1999) On the logic foundation of fuzzy reasoning. Inf Sci 117(1):47–88

    Article  MathSciNet  Google Scholar 

  • Wang L.X(1997) A course in fuzzy systems and control, Prentice Hall PTR, Upper Saddle River NJ07458,

  • Yager R (2004) On some new classes of implication operators and their role in approximate reasoning. Inf Sci 167:193–216

    Article  MathSciNet  Google Scholar 

  • Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning-I. Inf Sci 8:199–249

    Article  MathSciNet  Google Scholar 

  • Zeng XJ, Singh MG (1995) Approximation theory of fuzzy systems-MIMO case. IEEE Trans Fuzzy Syst 3(2):219–235

    Article  Google Scholar 

  • Zhou BK, Xu GQ, Li SJ (2015) The Quintuple implication principle of fuzzy reasoning. Inf Sci 297:202–215

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referees and the Editor-in-Chief for their valuable comments. This work was supported by the National Natural Science Foundation of China (Grant No. 61673352).

Author information

Authors and Affiliations

  1. School of Information and Engineering, Zhejiang Ocean University, Zhoushan, 316000, China

    Dechao Li

  2. Key Laboratory of Oceanographic Big Data Mining and Application of Zhejiang Province, Zhejiang Ocean University, Zhoushan, 316022, China

    Qingxue Zeng

Authors
  1. Dechao Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qingxue Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dechao Li.

Ethics declarations

Conflict of interest

Author declares that he has no conflict of interest.

Human and animal rights

This article does not contain any studies with human participants or animals performed by the authors.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, D., Zeng, Q. Approximate reasoning with aggregation functions satisfying GMP rules. Artif Intell Rev 55, 5575–5595 (2022). https://doi.org/10.1007/s10462-022-10136-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10462-022-10136-1

Keywords

Search

Navigation