Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Putting Rough Sets and Fuzzy Sets Together

  • Chapter
Intelligent Decision Support

Part of the book series: Theory and Decision Library ((TDLD,volume 11))

Abstract

In this paper we argue that fuzzy sets and rough sets aim to different purposes and that it is more natural to try to combine the two models of uncertainty (vagueness for fuzzy sets and coarseness for rough sets) in order to get a more accurate account of imperfect information. First, the upper and lower approximations of a fuzzy set are defined, when the universe of discourse of a fuzzy sets is coarsened by means of an equivalence relation. We then come close to Caianiello’s C-calculus. Shafer’s concept of coarsened belief functions also belongs to the same line of thought and is reviewed here. Another idea is to turn the equivalence relation relation into a fuzzy similarity relation, for a more expressive modeling of coarseness. New results on the representation of similarity relations by means of a fuzzy partition of fuzzy clusters of more or less indiscernible points are surveyed. The properties of upper and lower approximations of fuzzy sets by similarity relations are thoroughly studied. Lastly the potential usefulness of the fuzzy rough set notions for logical inference in the presence of both fuzzy predicates and graded indiscernibility is indicated. Especially fuzzy rough sets may provide a nice semantic background for modal logic involving fuzzy modalities and/or fuzzy sentences.

This paper draws from and continues a previous article by the authors, entitled “Rough fuzzy sets and fuzzy rough sets”, that appeared in the Int.J.of General Systems in 1990.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 259.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 329.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  • Bezdek, J.C. (1981) Pattern Recognition with Fuzzy Objective Function Algorithms, Plenum Press, New York.

    Book  MATH  Google Scholar 

  • Bezdek, J.C. and Harris, J.D. (1978) ‘Fuzzy partitions and relations — An axiomatic basis for clustering’, Fuzzy Sets and Systems 1, 112–127.

    MathSciNet  Google Scholar 

  • Bonissone, P.P. (1979) ‘A pattern recognition approach to the problem of linguistic approximation in system analysis’, in Proc. IEEE Inter. Conf. on Cybernetics and Society, Denver, pp. 793–798.

    Google Scholar 

  • Buckles, B.P. and Petry, F.E. (1982) ‘A fuzzy representation of data for relational databases’, Fuzzy Sets and Systems 7, 213–226.

    Article  MATH  Google Scholar 

  • Caianiello, E.R. (1973) ‘A calculus for hierarchical systems’, in Proc. 1st Inter. Conf. on Pattern Recognition, Washington, D.C., pp. 1–5.

    Google Scholar 

  • Caianiello, E.R. (1987) ‘C-calculus: an overview’, in E.R. Caianiello and M.A. Aizerman (eds.), Topics in the General Theory of Structures, D. Reidel Dordrecht, pp. 163–173.

    Google Scholar 

  • Caianiello, E.R. and Ventre, A.G.S. (1984) ‘C-calculus and fuzzy sets’, in Proc. 1st Napoli Meeting on the Mathematics of Fuzzy Systems, Università degli Studi di Napoli, pp. 29–33.

    Google Scholar 

  • Caianiello, E.R. and Ventre, A.G.S. (1985) ‘A model for C-calculus’, Int. J. of General Systems 11, 153–161.

    Google Scholar 

  • Chellas, B.F. (1980) Modal Logic: An Introduction, Cambridge University Press, Cambridge, UK.

    Google Scholar 

  • Dempster, A.P. (1967) ‘Upper and lower probabilities induced by a multiple-valued mapping’, Annals of Mathematical Statistics 38, 325–339.

    Article  MathSciNet  MATH  Google Scholar 

  • Di Nola, A., Sessa, A., Pedrycz, W. and Sanchez, E. (1989) Fuzzy Relation Equations and their Applications to Knowledge Engineering, Kluwer Academic Publ., Dordrecht, The Netherland.

    Google Scholar 

  • Dubois, D. and Prade, H. (1980) Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York.

    MATH  Google Scholar 

  • Dubois, D. and Prade, H. (1984a) ‘Fuzzy logics and the generalized modus ponens revisited’, Int. J. of Cybernetics and Systems 15, 293–331.

    Google Scholar 

  • Dubois, D. and Prade, H. (1984b) ‘A theorem on implication functions defined from triangular norms’, Stochastica 8 (3), 267–279.

    MathSciNet  MATH  Google Scholar 

  • Dubois, D. and Prade, H. (1985) ‘Evidence measures based on fuzzy information’, Automatica 21, 547–562.

    Article  MathSciNet  MATH  Google Scholar 

  • Dubois, D. and Prade, H. (1987) ‘Twofold fuzzy sets and rough sets — Some issues in knowledge representation’, Fuzzy Sets and Systems 23, 3–18.

    Article  MathSciNet  MATH  Google Scholar 

  • Dubois, D. and Prade, H. (with the collaboration of Farreny, H., Martin-Clouaire, R and Testemale, C.) (1988) Possibility Theory: An Approach to Computerized Processing of Uncertainty, Plenum Press, New York.

    Google Scholar 

  • Dubois, D. and Prade, H. (1990) ‘Rough fuzzy sets and fuzzy rough sets’, Int. J. of General Systems 17, 191–200.

    Google Scholar 

  • Dubois, D. and Prade, H. (1991) ‘Possibilistic logic, preferential models, non-monotonicity and related issues’, in Proc. of the 12th Inter. Joint Conf. on Artificial Intelligence (IJCAI-91), Sydney, Australia, Aug. 24–30, pp. 419–424.

    Google Scholar 

  • Dubois, D., Prade, H. and Testemale, C. (1988) In search of a modal system for possibility theory’, in Proc. of the Conf. on Artificial Intelligence, Munich, Germany, Aug. 1–5, pp. 501–506.

    Google Scholar 

  • Farinas del Cerro, L. and Herzig, A. (1991) ‘A modal analysis of possibility theory’, in Proc. of the Inter. Workshop on Fundamentals of Artificial Intelligence (FAIR’91), Smolenice Castle, Czechoslovakia, Sept. 8–12, 1991, Ph. Jorrand and J. Kelemen (eds.), Springer Verlag, Berlin, pp. 11–18.

    Google Scholar 

  • Farinas del Cerro, L. and Orlowska, E. (1985) ‘DAL — A logic for data analysis’, Theoretical Computer Science 36, 251–264.

    Article  MathSciNet  MATH  Google Scholar 

  • Farinas del Cerro, L. and Prade, H. (1986) ‘Rough sets, twofold fuzzy sets and modal logic —Fuzziness in indiscernibility and partial information’, in A. Di Nola and A.G.S. Ventre (eds.), The Mathematics of Fuzzy Systems, Verlag TÜV Rheinland, Köln, pp. 103–120.

    Google Scholar 

  • Fodor, J.C. (1991) ‘On fuzzy implication operators’, Fuzzy Sets and Systems 42, 293–300.

    Article  MathSciNet  MATH  Google Scholar 

  • Gisolfi, A. (1992) ‘An algebraic fuzzy structure for approximate reasoning’, Fuzzy Sets and Systems 45, 37–43.

    Article  MathSciNet  MATH  Google Scholar 

  • Höhle, U. (1988) ‘Quotients with respect to similarity relations’, Fuzzy Sets and Systems 27, 31–44.

    Article  MathSciNet  MATH  Google Scholar 

  • Kitainik, L. (1991) ‘Notes on convex decomposition of bold fuzzy equivalence relations’, in BUSEFAL n° 48 ( IRIT, Univ. P. Sabatier, Toulouse, France ), 27–35.

    Google Scholar 

  • Lewis, D. (1973) Counterfactuals, Blackwell, Oxford.

    Google Scholar 

  • López de Mdntaras, R. and Valverde, L. (1988) ‘New results in fuzzy clustering based on the concept of indistinguishability relation’, IEEE Trans. on Pattern Analysis and Machine Intelligence 10, 754–757.

    Article  Google Scholar 

  • Nakamura, A. (1988) ‘Fuzzy rough sets’, Note on Multiple-Valued Logic in Japan 9 (8), 1–8.

    Google Scholar 

  • Nakamura, A. (1989) ‘On a KTB-modal fuzzy logic’, Tech. Report n° C-31, Dept. of Applied Mathematics, Hiroshima University, Japan.

    Google Scholar 

  • Nakamura, A. (1991a) ‘Topological soft algebra for the S5 modal fuzzy logic’, in Proc. of the 21st Inter. Symp. on Multiple-Valued Logic, Victoria, B.C., pp. 80–84.

    Google Scholar 

  • Nakamura, A. (1991b) ‘On a logic based on fuzzy modalities’, Report MCS-10, Dept. of Computer Science, Meiji University, Japan.

    Google Scholar 

  • Nakamura, A. and Gao, J.M. (1991) ‘A logic for fuzzy data analysis’, Fuzzy Sets and Systems 39, 127–132.

    Article  MathSciNet  MATH  Google Scholar 

  • Novak, V. (1987) ‘Automatic generation of verbal comments on results of mathematical modelling’, in E. Sanchez and L.A. Zadeh (eds.), Approximate Reasoning in Intelligent Systems, Decision and Control, Pergamon Press, Oxford, U.K., pp. 55–68.

    Google Scholar 

  • Orlowska, E. (1984) ‘Modal logics in the theory of information systems’, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 30 (3), 213–222.

    Article  MathSciNet  MATH  Google Scholar 

  • Ovchinnikov, S.V. (1982)’On fuzzy relational systems’, in Proc. 2d World Conf. on Mathematics at the Service of Man, Las Palmas, Spain, pp. 566–569.

    Google Scholar 

  • Pawlak, Z. (1982) ‘Rough sets’, Int. J. of Computer and Information Science 11, 341–356.

    Google Scholar 

  • Pawlak, Z. (1984) ‘Rough classification’, Int. J. of Man-Machine Studies 20, 469–485.

    Google Scholar 

  • Pawlak, Z. (1985) ‘Rough sets and fuzzy sets’, Fuzzy Sets and Systems 17, 99–102.

    Article  MathSciNet  MATH  Google Scholar 

  • Pawlak, Z., Slowinski, K. and Slowinski, R. (1986) ‘Rough classification of patients after highly selective vagotomy for duodenal ulcer’, Int. J. of Man-Machine Studies 24, 413–433.

    Google Scholar 

  • Pawlak, Z., Wong, S.M.K. and Ziarko, W. (1988) ‘Rough sets: probabilistic versus deterministic approach’, Int. J. of Man-Machine Studies 29, 81–95.

    Google Scholar 

  • Pearl, J. and Verma, T. (1987) ‘The logic of representing dependencies by directed graphs’, in Proc. 6th AAAI National Conf. on Artificial Intelligence, Seattle, pp. 374–379.

    Google Scholar 

  • Pedrycz, W. (1985) ‘On generalized fuzzy relational equations and their applications’, J. of Mathematical Analysis and Applications 107, 520–536.

    Article  MathSciNet  MATH  Google Scholar 

  • Prade, H. (1984) ‘Lipskì’s approach to incomplete information data bases restated and generalized in the setting of Zadeh’s possibility theory’, Information Systems 9 (1), 27–42.

    Article  MathSciNet  MATH  Google Scholar 

  • Prade, H. and Testemale, C. (1984) ‘Generalizing database relational algebra for the treatment of uncertain/imprecice information and vague queries’, Information Sciences 34, 27–42.

    Article  MathSciNet  Google Scholar 

  • Prade, H. and Testemale, C. (1987) ‘Fuzzy relational databases: representational issues and reduction using similarity measures’, J. of the American Society for Information Science 38 (2), 118–126.

    Article  Google Scholar 

  • Roy, B. (1985) Méthodologie Multicritère d’Aide à la Décision, Editions Economica, Paris.

    Google Scholar 

  • Ruspini, E.H. (1991) ‘On the semantics of fuzzy logic’, Int. J. of Approximate Reasoning 5, 45–88.

    Google Scholar 

  • Sanchez, E. (1976) ‘Resolution of composite fuzzy relation equations’, Information and Control, 30 38–48

    Article  MathSciNet  MATH  Google Scholar 

  • Sanchez, E. (1978) ‘Resolution of eigen fuzzy sets equations’, Fuzzy Sets and Systems 1 (1), 69–74.

    Article  MathSciNet  MATH  Google Scholar 

  • Schweizer, B. and Sklar, A. (1983) Probabilistic Metric Spaces, North-Holland, Amsterdam.

    MATH  Google Scholar 

  • Shafer, G. (1976) A Mathematical Theory of Evidence, Princeton Univ. Press, Princeton, NJ.

    Google Scholar 

  • Shafer, G., Shennoy, P. and Mellouli, K. (1987) ‘Qualitative Markov trees’, Int. J. Approximate Reasoning 1, 349–400.

    Google Scholar 

  • Sugeno, M. (1977) ‘Fuzzy measures and fuzzy integrals: a survey’, in M.M. Gupta, G.N. Saridis and B.R. Gaines (eds.), Fuzzy Automated and Decision Processes, North-Holland, Amsterdam, pp. 89–102.

    Google Scholar 

  • Tahani, V. (1977) ‘A conceptual framework for fuzzy query processing — A step towards very intelligent database systems’, Information Processing and Management 13, 289–303.

    Article  MATH  Google Scholar 

  • Trillas, E. and Valverde, L. (1984) ‘An investigation into indisguishability operators’, in H.J. Skala, S. Termini and E. Trillas (eds.), Aspects of Vagueness, D. Reidel Dordrecht, pp. 231–256.

    Google Scholar 

  • Trillas, E. and Valverde, L. (1985) ‘On implication and indistinguishability in the setting of fuzzy logic’, in J. Kacprzyk and R.R. Yager (eds.), Management Decision Support Systems using Fuzzy Sets and Possibility Theory, Interdisciplinary Systems Research, Vol. 83, Verlag TÜV Rheinland, Köln, pp. 198–212.

    Google Scholar 

  • Valverde, L. (1985) ‘On the structure of F-indistinguishability operators’, Fuzzy Sets and Systems 17, 313–328.

    Article  MathSciNet  MATH  Google Scholar 

  • Willaeys, D. and Malvache, N. (1981) ‘The use of fuzzy sets for the treatment of fuzzy information by computer’, Fuzzy Sets and Systems 5, 323–328.

    Article  MATH  Google Scholar 

  • Wong, S.M.K. and Ziarko, W. (1985) ‘A probabilistic model of approximate classificaton and decision rules with uncertainties and inductive learning’, Technical Report CS 85–23, University of Regina, Saskatchewan.

    Google Scholar 

  • Wygralak, M. (1989) ‘Rough sets and fuzzy sets: some remarks on interrelations’, Fuzzy sets and Systems 29, 241–243.

    Article  MathSciNet  MATH  Google Scholar 

  • Zadeh, L.A. (1965) ‘Fuzzy sets’, Information and Control 8, 338–353.

    Article  MathSciNet  MATH  Google Scholar 

  • Zadeh, L.A. (1971) ‘Similarity relations and fuzzy orderings’, Information Sciences 3, 177–200.

    Article  MathSciNet  MATH  Google Scholar 

  • Zadeh, L.A. (1978) ‘Fuzzy sets as a basis for a theory of possibility’, Fuzzy Sets and Systems 1, 3–28.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut de Recherche en Informatique de Toulouse — C.N.R.S., Université Paul Sabatier, 31062, Toulouse, France

    Didier Dubois & Henri Prade

Authors
  1. Didier Dubois
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Henri Prade
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Computing Science, Technical University of Poznań, Poland

    Roman Słowiński

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Dubois, D., Prade, H. (1992). Putting Rough Sets and Fuzzy Sets Together. In: Słowiński, R. (eds) Intelligent Decision Support. Theory and Decision Library, vol 11. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7975-9_14

Download citation

  • DOI: https://doi.org/10.1007/978-94-015-7975-9_14

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4194-4

  • Online ISBN: 978-94-015-7975-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation