Algebras for Information Systems

Part of the Intelligent Systems Reference Library book series (ISRL, volume 42)


We present algebraic formalisms for different kinds of information systems, viz. deterministic, incomplete, and non-deterministic. Algebraic structures generated from these information systems are considered and corresponding abstract algebras are proposed. Representation theorems for these classes of abstract algebras are proved, which lead us to equational logics for deterministic, incomplete, and non-deterministic information systems.


Information system indiscernibility relation similarity relation Boolean algebra with operators 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Balbiani, P.: Axiomatization of logics based on Kripke models with relative accessibility relations. In: Orłowska, E. (ed.) Incomplete Information: Rough Set Analysis, pp. 553–578. Physica Verlag, Heidelberg (1998)Google Scholar
  2. 2.
    Balbiani, P., Orłowska, E.: A hierarchy of modal logics with relative accessibility relations. Journal of Applied Non-Classical Logics 9(2-3), 303–328 (1999)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Balbiani, P., Vakarelov, D.: A modal logic for indiscernibility and complementarity in information systems. Fundamenta Informaticae 50, 243–263 (1950)Google Scholar
  4. 4.
    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, Berlin (2004)CrossRefGoogle Scholar
  5. 5.
    Banerjee, M., Khan, M. A.: Propositional Logics from Rough Set Theory. In: Peters, J.F., Skowron, A., Düntsch, I., Grzymała-Busse, J.W., Orłowska, E., Polkowski, L. (eds.) Transactions on Rough Sets VI, Part I. LNCS, vol. 4374, pp. 1–25. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  6. 6.
    Birkhoff, G.: On the structure of abstract algebras. Proceedings of the Cambridge Philosophical Society 29, 441–464 (1935)CrossRefGoogle Scholar
  7. 7.
    Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press (2001)Google Scholar
  8. 8.
    Comer, S.: An algebraic approach to the approximation of information. Fundamenta Informaticae XIV, 492–502 (1991)MathSciNetGoogle Scholar
  9. 9.
    Düntsch, I., Gediga, G., Orłowska, E.: Relational attribute systems. International Journal of Human Computer Studies 55(3), 293–309 (2001)MATHCrossRefGoogle Scholar
  10. 10.
    Düntsch, I., Gediga, G., Orłowska, E.: Relational Attribute Systems II: Reasoning with Relations in Information Structures. In: Peters, J.F., Skowron, A., Marek, V.W., Orłowska, E., Słowiński, R., Ziarko, W.P. (eds.) Transactions on Rough Sets VII, Part II. LNCS, vol. 4400, pp. 16–35. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  11. 11.
    Demri, S., Orłowska, E.: Incomplete Information: Structure, Inference, Complexity. Springer, Heidelberg (2002)Google Scholar
  12. 12.
    Dubois, D., Prade, H.: Rough fuzzy sets and fuzzy rough sets. International Journal of General Systems 17, 191–200 (1990)MATHCrossRefGoogle Scholar
  13. 13.
    Henkin, L., Monk, J.D., Tarski, A.: Cylindric Algebras, Part I. North-Holland Pub. Co., Amsterdam (1971)Google Scholar
  14. 14.
    Iwiński, T.B.: Algebraic approach to rough sets. Bulletin of the Polish Academy of Sciences (Math) 35(9-10), 673–683 (1987)MATHGoogle Scholar
  15. 15.
    Khan, M. A., Banerjee, M.: A Logic for Complete Information Systems. In: Sossai, C., Chemello, G. (eds.) ECSQARU 2009. LNCS(LNAI), vol. 5590, pp. 829–840. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  16. 16.
    Khan, M.A., Banerjee, M.: Logics for information systems and their dynamic extensions. ACM Transactions on Computational Logic 12(4), 29 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Komorowski, J., Pawlak, Z., Polkowski, L., Skowron, A.: 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
  18. 18.
    Konikowska, B.: A formal language for reasoning about indiscernibility. Bulletin of the Polish Academy of Sciences 35, 239–249 (1987)MathSciNetMATHGoogle Scholar
  19. 19.
    Konikowska, B.: A logic for reasoning about relative similarity. Studia Logica 58, 185–226 (1997)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Kryszkiewicz, M.: Rough set approach to incomplete information systems. Information Sciences 112, 39–49 (1998)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Kryszkiewicz, M.: Rules in incomplete information systems. Information Sciences 113, 271–292 (1999)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Lin, T.Y., Yao, Y.Y.: Neighborhoods system: measure, probability and belief functions. In: Tsumoto, S., Kobayashi, S., Yokomori, T., Tanaka, H., Nakamura, A. (eds.) Proceedings of the The Fourth International Workshop on Rough Sets, Fuzzy Sets and Machine Discovery, November 6-8, pp. 202–207. The University of Tokyo, Tokyo (1996)Google Scholar
  23. 23.
    Orłowska, E.: Dynamic information system. Fundamenta Informaticae 5, 101–118 (1982)MathSciNetMATHGoogle Scholar
  24. 24.
    Orłowska, E.: Logic of Indiscernibility Relations. In: Skowron, A. (ed.) SCT 1984. LNCS, vol. 208, pp. 177–186. Springer, Heidelberg (1985)CrossRefGoogle Scholar
  25. 25.
    Orłowska, E.: Logic of nondeterministic information. Studia Logica 1, 91–100 (1985)CrossRefGoogle Scholar
  26. 26.
    Orłowska, E.: Kripke semantics for knowledge representation logics. Studia Logica XLIX, 255–272 (1990)CrossRefGoogle Scholar
  27. 27.
    Orłowska, E.: Rough set semantics for non-classical logics. In: Ziarko, W. (ed.) Rough Sets,Fuzzy Sets and Knowledge Discovery, Proceedings of the International Workshop on Rough Sets and Knowledge Discovery (RSKD 1993), Workshops in Computing, Banff, Alberta, Canada, October 12-15, pp. 143–148. Springer, Heidelberg (1993)Google Scholar
  28. 28.
    Orłowska, E., Pawlak, Z.: Representation of nondeterministic information. Theoretical Computer Science 29, 27–39 (1984)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Pawlak, Z.: Rough sets. International Journal of Computer and Information Science 11(5), 341–356 (1982)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Pawlak, Z.: Rough Sets. Theoretical Aspects of Reasoning about Data. Kluwer Academic Publishers, Dordrecht (1991)MATHCrossRefGoogle Scholar
  31. 31.
    Polkowski, L.: Rough Sets: Mathematical Foundations. Physica-Verlag, Heidelberg (2002)Google Scholar
  32. 32.
    Pomykała, J.A.: Approximation, similarity and rough constructions. ILLC prepublication series for computation and complexity theory CT-93-07, University of Amsderdam (1993)Google Scholar
  33. 33.
    Skowron, A., Stepaniuk, J.: Tolerance approximation spaces. Fundamenta Informaticae 27, 245–253 (1996)MathSciNetMATHGoogle Scholar
  34. 34.
    Ślęzak, D., Ziarko, W.: The investigation of the Bayesian rough set model. International Journal of Approximate Reasoning 40, 81–91 (2005)MathSciNetCrossRefGoogle Scholar
  35. 35.
    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, pp. 255–260. North–Holland (1987)Google Scholar
  36. 36.
    Vakarelov, D.: Modal logics for knowledge representation systems. Theoretical Computer Science 90, 433–456 (1991)MathSciNetMATHGoogle Scholar
  37. 37.
    Wasilewski, L., Ślęzak, D.: Foundations of rough sets from vagueness perspective. In: Hassanien, A.E., Suraj, Z., Ślęzak, D., Lingras, P. (eds.) Rough Computing, Theories, Technologies and Applications, pp. 1–37. IGI Global, Hershey (2008)Google Scholar
  38. 38.
    Yao, Y.Y.: Generalized rough set models. In: Polkowski, L., Skowron, A. (eds.) Rough Sets in Knowledge Discovery, pp. 286–318. Physica-Verlag, Heidelberg (1998)Google Scholar
  39. 39.
    Ziarko, W.: Variable precision rough set model. Journal of Computer and System Sciences 46, 39–59 (1993)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.The Institute of Mathematical SciencesChennaiIndia
  2. 2.Institute of MathematicsThe University of WarsawWarsawPoland
  3. 3.Indian Institute of TechnologyKanpurIndia

Personalised recommendations