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Fuzzy decision implication canonical basis

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Abstract

Fuzzy decision implication (FDI) is regarded as a basic form of knowledge representation in fuzzy decision based formal concept analysis. How to reduce redundant FDIs and generate an informative and minimal set of FDIs from a given set of FDIs is the main concern in the study of FDI. This paper introduces fuzzy decision premise, constructs fuzzy decision implication canonical basis (FD canonical basis) and proves that FD canonical basis is complete, non-redundant and optimal, i.e., FD canonical basis contains the least number of FDIs among all complete sets of FDIs. Thus, from a given set of FDIs, one can generate its corresponding FD canonical basis, which turns out to be informative (complete) and minimal (optimal).

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References

  1. Baczynski M, Jayaram B (2008) Fuzzy implications. studies in fuzziness and soft computing. Springer, Berlin, Heidelberg

    MATH  Google Scholar 

  2. Belohlávek R, Vychodil V (2005) Fuzzy equational logic, studies in fuzziness and soft computing, vol 186. Springer, Berlin, Heidelberg

    MATH  Google Scholar 

  3. Belohlavek R, Vychodil V (2006) Attribute implications in a fuzzy setting. In: ICFCA, pp 45–60

  4. Belohlávek R, Vychodil V (2006) Axiomatization of fuzzy attribute logic over complete residuated lattices. In: JCIS

  5. Belohlavek R, VychodiL V (2006) Fuzzy attribute logic over complete residuated lattices. J Exp Theor Artif Intell 18(4):471–480

    Article  Google Scholar 

  6. Belohlávek R, Vychodil V (2011) Basic algorithm for attribute implications and functional dependencies in graded setting. Int J Found Comput Sci 19(2):297–317

    Article  MathSciNet  Google Scholar 

  7. Belohlavek R, Vychodil V (2012) Formal concept analysis and linguistic hedges. Int J Gen Syst 41(5):503–532

    Article  MathSciNet  Google Scholar 

  8. Carpineto C, Romano G (2004) Concept data analysis: theory and applications. Wiley, Hoboken

    Book  Google Scholar 

  9. Ganter B, Wille R (1999) Formal concept analysis: mathematical foundations. Springer, Berlin

    Book  Google Scholar 

  10. Glodeanu CV (2011) Fuzzy-valued triadic implications. In: Napoli A, Vychodil V (eds) Proceedings of the eighth international conference on concept lattices and their applications, concept lattices and their applications, pp 159–173

  11. Guigues JL, Duquenne V (1986) Famille minimale d’implications informatives re ’sultant d’un tableau de donn’ees binaires. Math Sci Hum 24(95):5–18

    Google Scholar 

  12. Hájek P (1998) Metamathematics of fuzzy logic. Kluwer, Dordrecht

    MATH  Google Scholar 

  13. Hájek P (2001) On very true. Fuzzy Sets Syst 124(3):329–333

    Article  MathSciNet  Google Scholar 

  14. Hu K, Lu Y, Shi C (2000) An integrated mining approach for classification and association rule based on concept lattice. J Softwre 11(11):1479–1484

    Google Scholar 

  15. Ishigure H, Mutoh A, Matsui T et al (2015) Concept lattice reduction using attribute inference. In: Global conference on consumer electronics. IEEE, pp 108–111

  16. Kumar CA (2012) Fuzzy clustering-based formal concept analysis for association rules mining. Appl Artif Intell 26(3):274–301

    Article  Google Scholar 

  17. Kumar CA, Dias SM, J VN (2015) Knowledge reduction in formal contexts using non-negative matrix factorization. Math Comput Simul 109:46–63

    Article  MathSciNet  Google Scholar 

  18. Kumar CA, Srinivas S (2010) Concept lattice reduction using fuzzy K-Means clustering. Expert Syst Appl 37:2696–2704

    Article  Google Scholar 

  19. Li D, Zhang S, Zhai Y (2017) Method for generating decision implication canonical basis based on true premises. Int J Mach Learn Cybern 8(1):57–67

    Article  Google Scholar 

  20. Li J, Kumar CA, Mei C, Wang X (2017) Comparison of reduction in formal decision contexts. Int J Approx Reason 80:100–122

    Article  MathSciNet  Google Scholar 

  21. Li J, Mei C, Kumar CA, Zhang X (2013) On rule acquisition in decision formal contexts. Int J Mach Learn Cybern 4(6):721–731

    Article  Google Scholar 

  22. Li J, Mei C, Lv Y (2011) Knowledge reduction in decision formal contexts. Knowl Based Syst 24(5):709–715

    Article  Google Scholar 

  23. Li J, Mei C, Lv Y (2012) Knowledge reduction in formal decision contexts based on an order-preserving mapping. Int J Gen Syst 41(2):143–161

    Article  MathSciNet  Google Scholar 

  24. Li J, Mei C, Lv Y (2012) Knowledge reduction in real decision formal contexts. Inf Sci 189:191–207

    Article  MathSciNet  Google Scholar 

  25. Li J, Mei C, Lv Y (2013) Incomplete decision contexts: approximate concept construction, rule acquisition and knowledge reduction. Int J Approx Reason 54(1):149–165

    Article  MathSciNet  Google Scholar 

  26. Li J, Mei C, Xu W (2015) Concept learning via granular computing:a cognitive viewpoint. Inf Sci 298:447–467

    Article  MathSciNet  Google Scholar 

  27. Ma Y, Zhang X, Chi C (2011) Compact dependencies and intent waned values. J Softw 22(05):962–971

    Article  MathSciNet  Google Scholar 

  28. Qi J, Qian T, Wei L (2016) The connections between three-way and classical concept lattices. Knowl Based Syst 91:143–151

    Article  Google Scholar 

  29. Qi J, Wei L, Yao Y (2014) Three-way formal concept analysis. In: Miao D, Pedrycz W, Ślȩzak D, Peters G, Hu Q, Wang R (eds) Rough sets and knowledge technology: 9th international conference, RSKT 2014. Springer International Publishing, Shanghai, China, pp 732–741

    Google Scholar 

  30. Qu K, Zhai Y (2008) Generating complete set of implications for formal contexts. Knowl Based Syst 21(05):429–433

    Article  Google Scholar 

  31. Qu K, Zhai Y, Liang J, Chen M (2007) Study of decision implications based on formal concept analysis. Int J Gen Syst 36(02):147–156

    Article  MathSciNet  Google Scholar 

  32. Ren R, Wei L (2016) The attribute reductions of three-way concept lattices. Knowl Based Syst 99(C):92–102

    Article  Google Scholar 

  33. Roth C, Obiedkov S, Kourie D (2008) Towards concise representation for taxonomies of epistemic communities. In: Concept lattices and their applications. Springer, Berlin, Heidelberg, pp 240–255

  34. Sahami M (1995) Learning classification rules using lattices. In: Lecture notes in computer science. Springer, Berlin, Heidelberg, pp 343–346

    Google Scholar 

  35. Shivhare R, Cherukuri AK (2016) Three-way conceptual approach for cognitive memory functionalities. Int J Mach Learn Cybern 8(1):1–14

    Article  Google Scholar 

  36. Singh PK, Kumar CA, Gani A (2016) A comprehensive survey on formal concept analysis, its research trends and applications. Int J Appl Math Comput Sci 26(2):495–516

    Article  MathSciNet  Google Scholar 

  37. Tilley T, Cole R, Becker P, Eklund P (2005) A survey of formal concept analysis support for software engineering activities. In: Formal concept analysis. Springer, Berlin, Heidelberg, pp 250–271

    Chapter  Google Scholar 

  38. Wei L, Qi J, Zhang W (2008) Attribute reduction theory of concept lattice based on decision formal contexts. Sci Chin Ser F Inf Sci 51(7):910–923

    Article  MathSciNet  Google Scholar 

  39. Wille R (1982) Restructuring lattice theory: an approach based on hierarchies of concepts. In: Rival I (ed) Ordered sets, vol 83. Reidel, Dordrecht, pp 445–470

    Chapter  Google Scholar 

  40. Xia H, Chen Y, Gao H, Li Z, Chen Y (2010) Concept lattice-based semantic web service matchmaking. In: Communication software and networks, 2010. ICCSN’10. Second international conference on IEEE, pp 439–443

  41. Xu W, Li W (2016) Granular computing approach to two-way learning based on formal concept analysis in fuzzy datasets. IEEE Trans Cybern 46(2):366–379

    Article  MathSciNet  Google Scholar 

  42. Xu W, Pang J, Luo S (2014) A novel cognitive system model and approach to transformation of information granules. Int J Approx Reason 55(3):853–866

    Article  MathSciNet  Google Scholar 

  43. Zhai J, Zhang Y, Zhu H (2017) Three-way decisions model based on tolerance rough fuzzy set. Int J Mach Learn Cybern 8(1):35–43

    Article  Google Scholar 

  44. Zhai Y, Li D, Qu K (2013) Fuzzy decision implications. Knowl Based Syst 37:230–236

    Article  Google Scholar 

  45. Zhai Y, Li D, Qu K (2014) Decision implications: a logical point of view. Int J Mach Learn Cybern 5(4):509–516

    Article  Google Scholar 

  46. Zhai Y, Li D, Qu K (2015) Canonical basis for decision implications. Chin J Electron 43(1):18–23

    Google Scholar 

  47. Zhai Y, Li D, Qu K (2015) Decision implication canonical basis: a logical perspective. J Comput Syst Sci 81(1):208–218

    Article  MathSciNet  Google Scholar 

  48. Zhai YH, Li DY, Qu KS (2012) Probability fuzzy attribute implications for interval-valued fuzzy sets. Int J Database Theory Appl 5(4):95–108

    Google Scholar 

  49. Zhai YH, Li DY, Qu KS (2012) probability inclusion degrees and probability fuzzy attribute implications for interval-valued fuzzy sets. In: International conference, ISI 2012

  50. Zhi H, Li J (2016) Granule description based on formal concept analysis. Knowl Based Syst 104:62–73

    Article  Google Scholar 

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Acknowledgements

The works described in this paper are supported by National Natural Science Foundation of China (nos. 61672331,61303107, 61573231, 41401521, 61272095), the Project supported by the State Key Program of National Natural Science of China (nos. 61432011, U1435212), Shanxi Scholarship Council of China (2013-2014), the Natural Science Foundation of Shanxi, China (nos. 201601D021072, 2015021101), Project supported by National Science and Technology (no. 2012BAH33B01) and Shanxi Science and Technology Infrastructure (201601D021076, 2015091001-0102).

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Authors and Affiliations

  1. Key Laboratory of Computational Intelligence, Chinese Information Processing of Ministry of Education, Taiyuan, 030006, China

    Yanhui Zhai & Deyu Li

  2. School of Computer and Information Technology, Shanxi University, Taiyuan, 030006, China

    Yanhui Zhai, Deyu Li & Kaishe Qu

Authors
  1. Yanhui Zhai
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  2. Deyu Li
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  3. Kaishe Qu
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Correspondence to Deyu Li.

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Zhai, Y., Li, D. & Qu, K. Fuzzy decision implication canonical basis. Int. J. Mach. Learn. & Cyber. 9, 1909–1917 (2018). https://doi.org/10.1007/s13042-017-0780-7

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  • DOI: https://doi.org/10.1007/s13042-017-0780-7

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