Propositional Logics from Rough Set Theory

  • Mohua Banerjee
  • Md. Aquil Khan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4374)


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.


Modal Logic Multiagent System Propositional Logic Approximation Space Standard Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 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. 2.
    Pawlak, Z.: Rough Sets. Theoretical Aspects of Reasoning about Data. Kluwer Academic Publishers, Dordrecht (1991)zbMATHGoogle Scholar
  3. 3.
    Pawlak, Z.: Rough logic. Bull. Polish Acad. Sc (Tech. Sc.) 35, 253–258 (1987)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Orłowska, E.: Kripke semantics for knowledge representation logics. Studia Logica XLIX, 255–272 (1990)Google Scholar
  5. 5.
    Chakraborty, M.K., Banerjee, M.: Rough consequence. Bull. Polish Acad. Sc(Math.) 41(4), 299–304 (1993)MathSciNetzbMATHGoogle Scholar
  6. 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. 7.
    Banerjee, M., Chakraborty, M.K.: Rough sets through algebraic logic. Fundamenta Informaticae 28(3,4), 211–221 (1996)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Düntsch, I.: A logic for rough sets. Theoretical Computer Science 179, 427–436 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 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)CrossRefGoogle Scholar
  10. 10.
    Banerjee, M.: Logic for rough truth. Fundamenta Informaticae 71(2-3), 139–151 (2006)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Farinas Del Cerro, L., Orłowska, E.: DAL – a logic for data analysis. Theoretical Computer Science 36, 251–264 (1997)CrossRefGoogle Scholar
  12. 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) 255–260Google Scholar
  13. 13.
    Vakarelov, D.: A modal logic for similarity relations in Pawlak knowledge representation systems. Fundamenta Informaticae 15, 61–79 (1991)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Vakarelov, D.: Modal logics for knowledge representation systems. Theoretical Computer Science 90, 433–456 (1991)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Nakamura, A.: A rough logic based on incomplete information and its application. Int. J. Approximate Reasoning 15, 367–378 (1996)CrossRefzbMATHGoogle Scholar
  16. 16.
    Orłowska, E., Pawlak, Z.: Representation of nondeterministic information. Theoretical Computer Science 29, 27–39 (1984)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Vakarelov, D., Balbiani, P.: A modal logic for indiscernibilty and complementarity in information systems. Fundamenta Informaticae 50, 243–263 (2002)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Orłowska, E.: Logic of nondeterministic information. Studia Logica 1, 91–100 (1985)CrossRefGoogle Scholar
  19. 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. 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. 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. 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. 23.
    Polkowski, L., Skowron, A.: Rough mereology: a new paradigm for approximate reasoning. Int. J. Approximate Reasoning 15(4), 333–365 (1997)CrossRefMathSciNetGoogle Scholar
  24. 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. 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)CrossRefGoogle Scholar
  26. 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. 27.
    Yao, Y.: Constructive and algebraic methods of the theory of rough sets. Information Sciences 109, 21–47 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Pomykała, J.: Approximation, similarity and rough construction. Preprint CT–93–07, ILLC Prepublication Series, University of Amsterdam (1993)Google Scholar
  29. 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. 30.
    Rasiowa, H.: An Algebraic Approach to Non-classical Logics. North Holland, Amsterdam (1974)zbMATHGoogle Scholar
  31. 31.
    Banerjee, M.: Rough sets and 3-valued Łukasiewicz logic. Fundamenta Informaticae 32, 213–220 (1997)Google Scholar
  32. 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. 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. 34.
    Boicescu, V., et al.: Łukasiewicz-Moisil Algebras. North Holland, Amsterdam (1991)zbMATHGoogle Scholar
  35. 35.
    Comer, S.: Perfect extensions of regular double Stone algebras. Algebra Universalis 34, 96–109 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  36. 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. 37.
    Pawlak, Z.: Rough sets. Int. J. Comp. Inf. Sci. 11(5), 341–356 (1982)CrossRefMathSciNetzbMATHGoogle Scholar
  38. 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)CrossRefGoogle Scholar
  39. 39.
    Pawlak, Z.: Rough relations. ICS PAS Reports 435 (1981)Google Scholar
  40. 40.
    Skowron, A., Stepaniuk, J.: Tolerance approximation spaces. Fundamenta Informaticae 27, 245–253 (1996)MathSciNetzbMATHGoogle Scholar
  41. 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)CrossRefGoogle Scholar
  42. 42.
    da Costa, N.C.A., Doria, F.A.: On Jaśkowski’s discussive logics. Studia Logica 54, 33–60 (1995)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Mohua Banerjee
    • 1
  • Md. Aquil Khan
    • 1
  1. 1.Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur 208 016India

Personalised recommendations