Skip to main content
SpringerLink
Log in

Relationships between two pairs of covering approximation operators and belief structures

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

As two mathematical tools for dealing with uncertainty, the covering rough set theory and the evidence theory have close relationships with each other. Different covering rough set models are characterized by evidence theory. The purpose of this paper is to interpret belief functions with two pairs of covering approximation operators. The two pairs of covering approximation operators are equivalent to a pair of relation approximation operators. Then, based on a necessary and sufficient condition for a belief structure to be the belief structure induced by the pair of relation approximation operators, necessary and sufficient conditions for a belief structure to be the belief structure induced by the covering approximation operators are presented. Moreover, two kinds of covering reductions in covering information systems defined from the two pairs of covering approximation operators are characterized by the belief and plausibility functions.

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.

Similar content being viewed by others

References

  • Bonikowski Z, Bryniarski E, Wybraniec-Skardowska U (1998) Extensions and intentions in the rough set theory. Inf Sci 107:149–167

    Article  MathSciNet  MATH  Google Scholar 

  • Bryniarski E (1989) A calculus of rough sets of the first order. Bull Pol Acad Sci 36(16):71–77

    MathSciNet  MATH  Google Scholar 

  • Che X, Mi J (2019) Attributes set reduction in multigranulation approximation space of a multi-source decision information system. Int J Mach Learn Cyber 10:2297–2311

    Article  Google Scholar 

  • Chen D, Wang C, Hu Q (2007) A new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough sets. Inform Sci 177:3500–3518

    Article  MathSciNet  MATH  Google Scholar 

  • Chen D, Li W, Zhang X, Kwong S (2014) Evidence-theory-based numerical algorithms of attribute reduction with neighborhood-covering rough sets. Int J Approx Reason 55:908–923

    Article  MathSciNet  MATH  Google Scholar 

  • Chen D, Zhang X, Li W (2015) On measurements of covering rough sets based on granules and evidence theory. Inf Sci 317:329–348

    Article  MathSciNet  MATH  Google Scholar 

  • Dempster AP (1967) Upper and lower probabilities induced by a multivalued mapping. Ann Math Sta 38(2):325–339

    Article  MathSciNet  MATH  Google Scholar 

  • Feng T, Zhang S, Mi J (2012) The reduction and fusion of fuzzy covering systems based on the evidence theory. Int J Approx Reason 53:87–103

    Article  MathSciNet  MATH  Google Scholar 

  • Li J (2005) Rough sets and subsets of a topological Space. Syst Eng Theory Pract 7:136–140 ((in Chinese))

    Google Scholar 

  • Lin TY (1998) Granular computing on binary relations II: rough set representations and belief functions. In: Polkowski L, Skowron A (eds) Rough sets in knowledge discovery: 1. methodolodgy and applications study in fuzziness and soft computing, vol 18. Physica, Heidelberg, pp 122–140

    Google Scholar 

  • Lin G, Liang J, Qian Y (2015) An information fusion approach by combining multigranulation rough sets and evidence theory. Inf Sci 314:184–199

    Article  MathSciNet  MATH  Google Scholar 

  • Liu G (2013) The relationship among different covering approximations. Inf Sci 250:178–183

    Article  MathSciNet  MATH  Google Scholar 

  • Liu G, Sai Y (2009) A comparison of two types of rough sets induced by coverings. Int J Approx Reason 50:521–528

    Article  MathSciNet  MATH  Google Scholar 

  • Lu J, Li D, Zhai Y, Bai H (2019) Belief and plausibility functions of type-2 fuzzy rough sets. Int J Approx Reason 105:194–216

    Article  MathSciNet  MATH  Google Scholar 

  • Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11:341–356

    Article  MATH  Google Scholar 

  • Pomykala JA (1987) Approximation operations in approximation space. Bull Pol Acad Sci 35(9–10):653–662

    MathSciNet  MATH  Google Scholar 

  • Qin K, Gao Y, Pei Z (2007) On covering rough sets. In: The Second International Conference on Rough Sets and Knowledge Technology (RSKT 2007). Lect Notes Comput Sci, vol 4481, pp 34–41

  • Restrepo M, Cornelis C, Gómez J (2014) Partial order relation for approximation operators in covering based rough sets. Inf Sci 284:44–59

    Article  MathSciNet  MATH  Google Scholar 

  • Restrepo M, Cornelis C, Gómez J (2014) Duality, conjugacy and adjointness of approximation operators in covering-based rough sets. Int J Approx Reason 55:469–485

    Article  MathSciNet  MATH  Google Scholar 

  • Shafer G (1976) A mathematical theory of evidence. Princeton University Press, Princeton

    Book  MATH  Google Scholar 

  • Skowron A (1989) The relationship between rough set theory and evidence theory. Bull Pol Acad Sci Math 37:87–90

    Google Scholar 

  • Skowron A (1990) The rough sets theory and evidence theory. Fundam Inform 13:245–262

    Article  MathSciNet  MATH  Google Scholar 

  • Skowron A, Grzymala-Busse J (1994) From rough set theory to evidence theory. In: Yager RR, Fedrizzi M, Kacprzyk J (eds) Advance in the Dempster–Shafer theory of evidence. Wiley, NewYork, pp 193–236

    Google Scholar 

  • Tan A, Wu W, Li J, Lin G (2016) Evidence-theory-based numerical characterization of multigranulation rough sets in incomplete information systems. Fuzzy Sets Syst 294:18–35

    Article  MathSciNet  MATH  Google Scholar 

  • Tan A, Wu W, Tao Y (2017) On the belief structures and reductions of multigranulation spaces with decisions. Int J Approx Reason 88:39–52

    Article  MathSciNet  MATH  Google Scholar 

  • Tan A, Wu W, Tao Y (2018) A unified framework for characterizing rough sets with evidence theory in various approximation spaces. Inf Sci 454–455:144–160

    Article  MathSciNet  MATH  Google Scholar 

  • Tsang E, Cheng D, Lee J, Yeung D (2004) On the upper approximations of covering generalized rough sets. In: Proceedings of the 3rd international conference machine learning and cybernetics, pp 4200–4203

  • Wang L, Yang X, Yang J, Wu C (2012) Relationships among generalized rough sets in six coverings and pure reflexive neighborhood system. Inf Sci 207:66–78

    Article  MathSciNet  MATH  Google Scholar 

  • Wu W (2012) Knowledge reduction in random incomplete decision tables via evidence theory. Fundam Inform 115:203–218

    Article  MathSciNet  MATH  Google Scholar 

  • Wu W, Leung Y, Zhang W (2002) Connections between rough set theory and Dempster–Shafer theory of evidence. Int J Gen Syst 31:405–430

    Article  MathSciNet  MATH  Google Scholar 

  • Wu W, Zhang M, Li H, Mi J (2005) Knowledge reduction in random information systems via Dempster–Shafer theory of evidence. Inf Sci 174:143–164

    Article  MathSciNet  MATH  Google Scholar 

  • Wu W, Mi J, Li T (2012) Rough approximation spaces and belief structures in infinite universes of discourse. J Comput Res Dev 49(2):327–336 ((in Chinese))

    Google Scholar 

  • Xu Z, Wang Q (2005) On the properties of covering rough sets model. J Henan Normal Univ (Natural Sciences) 33(1):130–132 ((in Chinese))

    MathSciNet  MATH  Google Scholar 

  • Xu W, Zhang W (2007) Measuring roughness of generalized rough sets induced by a covering. Fuzzy Sets Syst 158:2443–2455

    Article  MathSciNet  MATH  Google Scholar 

  • Xu W, Zhang X, Zhong J, Zhang W (2010) Attribute reduction in ordered information systems based on evidence theory. Knowl Inf Syst 25:169–184

    Article  Google Scholar 

  • Yao Y (1998) Relational interpretations of neighborhood operators and rough set approximation operators. Inf Sci 111(1–4):239–259

    Article  MathSciNet  MATH  Google Scholar 

  • Yao Y (2003) On generalizing rough set theory. In: The 9th International Conference on Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing (RSFDGrc 2003). Lect Notes Comput Sci, vol 2639, pp 44–51

  • Yao Y, Lingras P (1998) Interpretations of belief functions in the theory of rough sets. Inf Sci 104:81–106

    Article  MathSciNet  MATH  Google Scholar 

  • Yao Y, Yao B (2012) Covering based rough set approximations. Inf Sci 200:91–107

    Article  MathSciNet  MATH  Google Scholar 

  • Yun Z, Ge X, Bai X (2011) Axiomatization and conditions for neighborhoods in a covering to form a partition. Inf Sci 181:1735–1740

    Article  MathSciNet  MATH  Google Scholar 

  • Żakowski W (1983) Approximations in the space \((U,\Pi )\). Demonstr Math 16:761–769

    MATH  Google Scholar 

  • Zhang Y, Li J (2010) The reduction of covering generalized rough sets. Fuzzy Syst Math 24(3):138–143 ((in Chinese))

    MathSciNet  MATH  Google Scholar 

  • Zhang Y, Li C (2020) Relationships between relation-based rough sets and belief structures. Int J Approx Reason 127:83–98

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang Y, Li C (2021) Numerical characterizations of topological reductions of covering information systems in evidence theory. Math Probl Eng 2021:1–9

    MathSciNet  Google Scholar 

  • Zhang Y, Luo M (2013) Relationships between covering-based rough sets and relation-based rough sets. Inf Sci 225:55–71

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang Y, Li J, Wu W (2010) On axiomatic characterizations of three pairs of covering-based approximation operators. Inf Sci 180:174–187

    Article  MathSciNet  Google Scholar 

  • Zhang Y, Li C, Li J (2019) On characterizations of a pair of covering-based approximation operators. J Soft Comput 23(12):3965–3972

    Article  MATH  Google Scholar 

  • Zhang S, Sun P, Mi J, Feng T (2020) Belief function of Pythagorean fuzzy rough approximation space and its applications. Int J Approx Reason 119:58–80

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao Z (2016) On some types of covering rough sets from topological points of view. Int J Approx Reason 68:1–14

    Article  MathSciNet  MATH  Google Scholar 

  • Zhu W (2007) Topological approaches to covering rough sets. Inf Sci 177:1499–1508

    Article  MathSciNet  MATH  Google Scholar 

  • Zhu W, Wang FY (2007) On three types of covering-based rough sets. IEEE Trans Knowl Data Engin 19(8):1131–1144

    Article  Google Scholar 

Download references

Funding

This work was supported by Grants from the National Natural Science Foundation of China (Nos. 11701258, 11871259), Natural Science Foundation of Fujian (nos. 2019J01749, 2020J01801, 2020J02043).

Author information

Authors and Affiliations

  1. College of Computer, Minnan Normal University, Zhangzhou, 363000, Fujian, China

    Yan-Lan Zhang

  2. School of Mathematics and Statistics, Minnan Normal University, Zhangzhou, 363000, Fujian, China

    Chang-Qing Li

  3. Key Laboratory of Data Science and Intelligence Application, Fujian Province University, Zhangzhou, 363000, China

    Yan-Lan Zhang

  4. Key Laboratory of Granular Computing, Minnan Normal University, Zhangzhou, 363000, China

    Chang-Qing Li

Authors
  1. Yan-Lan Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chang-Qing Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yan-Lan Zhang.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interests.

Ethical approval

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

Additional information

Communicated by Regivan Hugo Nunes Santiago.

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

Zhang, YL., Li, CQ. Relationships between two pairs of covering approximation operators and belief structures. Comp. Appl. Math. 41, 168 (2022). https://doi.org/10.1007/s40314-022-01848-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-022-01848-9

Keywords

Mathematics Subject Classification

Search

Navigation