Skip to main content
SpringerLink
Log in

Propositional Logics from Rough Set Theory

  • Chapter
Transactions on Rough Sets VI

Part of the book series: Lecture Notes in Computer Science ((TRS,volume 4374))

Abstract

The article focusses on propositional logics with semantics based on rough sets. Many approaches to rough sets (including generalizations) have come to the fore since the inception of the theory, and resulted in different “rough logics” as well. The essential idea behind these logics is, quite naturally, to interpret well-formed formulae as rough sets in (generalized) approximation spaces. The syntax, in most cases, consists of modal operators along with the standard Boolean connectives, in order to reflect the concepts of lower and upper approximations. Non-Boolean operators make appearances in some cases too.

Information systems (“complete” and “incomplete”) have always been the “practical” source for approximation spaces. Characterization theorems have established that a rough set semantics based on these “induced” spaces, is no different from the one mentioned above. We also outline some other logics related to rough sets, e.g. logics of information systems – which, in particular, feature expressions corresponding to attributes in their language. These systems address various issues, such as the temporal aspect of information, multiagent systems, rough relations.

An attempt is made here to place this gamut of work, spread over the last 20 years, in one platform. We present the various relationships that emerge and indicate questions that surface.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight 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.

References

  1. Banerjee, M., Chakraborty, M.K.: Algebras from rough sets. In: Pal, S.K., Polkowski, L., Skowron, A. (eds.) Rough-neuro Computing: Techniques for Computing with Words, pp. 157–184. Springer, Heidelberg (2004)

    Google Scholar 

  2. Pawlak, Z.: Rough Sets. Theoretical Aspects of Reasoning about Data. Kluwer Academic Publishers, Dordrecht (1991)

    MATH  Google Scholar 

  3. Pawlak, Z.: Rough logic. Bull. Polish Acad. Sc (Tech. Sc.) 35, 253–258 (1987)

    MathSciNet  MATH  Google Scholar 

  4. Orłowska, E.: Kripke semantics for knowledge representation logics. Studia Logica XLIX, 255–272 (1990)

    Google Scholar 

  5. Chakraborty, M.K., Banerjee, M.: Rough consequence. Bull. Polish Acad. Sc(Math.) 41(4), 299–304 (1993)

    MathSciNet  MATH  Google Scholar 

  6. Yao, Y., Lin, T.Y.: Generalization of rough sets using modal logics. Intelligent Automation and Soft Computing 2, 103–120 (1996)

    Google Scholar 

  7. Banerjee, M., Chakraborty, M.K.: Rough sets through algebraic logic. Fundamenta Informaticae 28(3,4), 211–221 (1996)

    MathSciNet  MATH  Google Scholar 

  8. Düntsch, I.: A logic for rough sets. Theoretical Computer Science 179, 427–436 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dai, J.H.: Logic for rough sets with rough double Stone algebraic semantics. In: Ślęzak, D., et al. (eds.) RSFDGrC 2005. LNCS (LNAI), vol. 3641, Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  10. Banerjee, M.: Logic for rough truth. Fundamenta Informaticae 71(2-3), 139–151 (2006)

    MathSciNet  MATH  Google Scholar 

  11. Farinas Del Cerro, L., Orłowska, E.: DAL – a logic for data analysis. Theoretical Computer Science 36, 251–264 (1997)

    Article  Google Scholar 

  12. Vakarelov, D.: Abstract characterization of some knowledge representation systems and the logic NIL of nondeterministic information. In Jorrand, P., Sgurev, V., eds.: Artificial Intelligence II. North-Holland (1987) 255–260

    Google Scholar 

  13. Vakarelov, D.: A modal logic for similarity relations in Pawlak knowledge representation systems. Fundamenta Informaticae 15, 61–79 (1991)

    MathSciNet  MATH  Google Scholar 

  14. Vakarelov, D.: Modal logics for knowledge representation systems. Theoretical Computer Science 90, 433–456 (1991)

    MathSciNet  MATH  Google Scholar 

  15. Nakamura, A.: A rough logic based on incomplete information and its application. Int. J. Approximate Reasoning 15, 367–378 (1996)

    Article  MATH  Google Scholar 

  16. Orłowska, E., Pawlak, Z.: Representation of nondeterministic information. Theoretical Computer Science 29, 27–39 (1984)

    Article  MathSciNet  Google Scholar 

  17. Vakarelov, D., Balbiani, P.: A modal logic for indiscernibilty and complementarity in information systems. Fundamenta Informaticae 50, 243–263 (2002)

    MathSciNet  MATH  Google Scholar 

  18. Orłowska, E.: Logic of nondeterministic information. Studia Logica 1, 91–100 (1985)

    Article  Google Scholar 

  19. Orłowska, E.: Logic of indiscernibility relations. In: Skowron, A. (ed.) Computation Theory. LNCS, vol. 208, pp. 177–186. Springer, Heidelberg (1985)

    Google Scholar 

  20. Rauszer, C.M.: Rough logic for multiagent systems. In: Masuch, M., Polos, L. (eds.) Knowledge Representation and Reasoning Under Uncertainty. LNCS, vol. 808, pp. 161–181. Springer, Heidelberg (1994)

    Google Scholar 

  21. Stepaniuk, J.: Rough relations and logics. In: Polkowski, L., Skowron, A. (eds.) Rough Sets in Knowledge Discovery 1: Methodology and Applications, pp. 248–260. Physica-Verlag, Heidelberg (1998)

    Google Scholar 

  22. Małuszyński, J., Vitória, A.: Toward rough datalog: embedding rough sets in prolog. In: Pal, S.K., Polkowski, L., Skowron, A. (eds.) Rough-neuro Computing: Techniques for Computing with Words, pp. 297–332. Springer, Berlin (2004)

    Google Scholar 

  23. Polkowski, L., Skowron, A.: Rough mereology: a new paradigm for approximate reasoning. Int. J. Approximate Reasoning 15(4), 333–365 (1997)

    Article  MathSciNet  Google Scholar 

  24. Wong, S.K.M.: A rough set model for reasoning about knowledge. In: Orłowska, E. (ed.) Incomplete Information: Rough Set Analysis. Studies in Fuzziness and Soft Computing, vol. 13, pp. 276–285. Physica-Verlag, Heidelberg (1998)

    Google Scholar 

  25. Banerjee, M.: Rough belief change. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets V. LNCS, vol. 4100, pp. 25–38. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  26. Banerjee, M., Chakraborty, M.K.: Rough consequence and rough algebra. In: Ziarko, W.P. (ed.) Rough Sets, Fuzzy Sets and Knowledge Discovery, Proc. Int. Workshop on Rough Sets and Knowledge Discovery (RSKD’93). Workshops in Computing, Springer, London (1994)

    Google Scholar 

  27. Yao, Y.: Constructive and algebraic methods of the theory of rough sets. Information Sciences 109, 21–47 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  28. Pomykała, J.: Approximation, similarity and rough construction. Preprint CT–93–07, ILLC Prepublication Series, University of Amsterdam (1993)

    Google Scholar 

  29. Komorowski, J., et al.: Rough sets: a tutorial. In: Pal, S.K., Skowron, A. (eds.) Rough Fuzzy Hybridization: A New Trend in Decision-Making, pp. 3–98. Springer, Singapore (1999)

    Google Scholar 

  30. Rasiowa, H.: An Algebraic Approach to Non-classical Logics. North Holland, Amsterdam (1974)

    MATH  Google Scholar 

  31. Banerjee, M.: Rough sets and 3-valued Łukasiewicz logic. Fundamenta Informaticae 32, 213–220 (1997)

    Google Scholar 

  32. Pagliani, P.: Rough set theory and logic-algebraic structures. In: Orłowska, E. (ed.) Incomplete Information: Rough Set Analysis. Studies in Fuzziness and Soft Computing, vol. 13, pp. 109–190. Physica-Verlag, Heidelberg (1998)

    Google Scholar 

  33. Iturrioz, L.: Rough sets and three-valued structures. In: Orłowska, E. (ed.) Logic at Work: Essays Dedicated to the Memory of Helena Rasiowa. Studies in Fuzziness and Soft Computing, vol. 24, pp. 596–603. Physica-Verlag, Heidelberg (1999)

    Google Scholar 

  34. Boicescu, V., et al.: Łukasiewicz-Moisil Algebras. North Holland, Amsterdam (1991)

    MATH  Google Scholar 

  35. Comer, S.: Perfect extensions of regular double Stone algebras. Algebra Universalis 34, 96–109 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  36. Andréka, H., Németi, I., Sain, I.: Abstract model theoretic approach to algebraic logic. CCSOM working paper, Department of Statistics and Methodology, University of Amsterdam (1992)

    Google Scholar 

  37. Pawlak, Z.: Rough sets. Int. J. Comp. Inf. Sci. 11(5), 341–356 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  38. Pagliani, P., Chakraborty, M.K.: Information quanta and approximation spaces I: non-classical approximation operators. In: Proc. 2005 IEEE Conf. on Granular Computing, pp. 605–610. IEEE Computer Society Press, Los Alamitos (2005)

    Chapter  Google Scholar 

  39. Pawlak, Z.: Rough relations. ICS PAS Reports 435 (1981)

    Google Scholar 

  40. Skowron, A., Stepaniuk, J.: Tolerance approximation spaces. Fundamenta Informaticae 27, 245–253 (1996)

    MathSciNet  MATH  Google Scholar 

  41. Polkowski, L.: Rough mereological reasoning in rough set theory: recent results and problems. In: Wang, G.-Y., et al. (eds.) RSKT 2006. LNCS (LNAI), vol. 4062, Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  42. da Costa, N.C.A., Doria, F.A.: On Jaśkowski’s discussive logics. Studia Logica 54, 33–60 (1995)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur 208 016, India

    Mohua Banerjee & Md. Aquil Khan

Authors
  1. Mohua Banerjee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Md. Aquil Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

James F. Peters Andrzej Skowron Ivo Düntsch Jerzy Grzymała-Busse Ewa Orłowska Lech Polkowski

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer Berlin Heidelberg

About this chapter

Cite this chapter

Banerjee, M., Khan, M.A. (2007). Propositional Logics from Rough Set Theory. In: Peters, J.F., Skowron, A., Düntsch, I., Grzymała-Busse, J., Orłowska, E., Polkowski, L. (eds) Transactions on Rough Sets VI. Lecture Notes in Computer Science, vol 4374. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71200-8_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-71200-8_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-71198-8

  • Online ISBN: 978-3-540-71200-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation