Knowledge Algebras and Their Discrete Duality

  • Ewa OrłowskaEmail author
  • Anna Maria Radzikowska
Part of the Intelligent Systems Reference Library book series (ISRL, volume 43)


A class of knowledge algebras inspired by a logic with the knowledge operator presented in [17] is introduced . Knowledge algebras provide a formalization of the Hintikka knowledge operator [8] and reflect its rough set semantics. A discrete duality is proved for the class of knowledge algebras and a corresponding class of knowledge frames.


Boolean algebra knowledge operator knowledge algebra knowledge frame rough set discrete duality representation theorem canonical frame complex algebra 


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  1. 1.
    van Benthem, J.: Correspondence theory. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Pholosophical Logic, vol. 2, pp. 167–247. D. Reidel, Dordrecht (1984)CrossRefGoogle Scholar
  2. 2.
    Chellas, B.F.: Modal Logic: An Introduction. Cambridge University Press, Cambridge (1980)zbMATHCrossRefGoogle Scholar
  3. 3.
    Davey, B.A., Priestley, H.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  4. 4.
    Demri, S.: A completeness proof for a logic with an alternative necessity operator. Studia Logica 58, 99–112 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Demri, S.: A logic with relative knowledge operators. Journal of Logic, Language and Information 8, 167–185 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Demri, S., Orłowska, E.: Incomplete Information: Structure, Inference, Complexity. EATCS Monographs in Teoretical Computer Science. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  7. 7.
    Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning about Knowledge. MIT Press, Cambridge (1995)zbMATHGoogle Scholar
  8. 8.
    Hintikka, J.: Knowledge and Belief. Cornell University Press, London (1962)Google Scholar
  9. 9.
    van der Hoek, W., Meyer, J.J.: A complete epistemic logic for multiple agents. In: Bacharach, M., Gérard-Varet, L.A., Mongin, P., Shin, H. (eds.) Epistemic Logic and the Theory of Games and Decisions, pp. 35–68. Kluwer Academic Publishers, Dordrecht (1999)Google Scholar
  10. 10.
    Höhle, U.: Commutative, residuated l–monoids. In: Höhle, U., Klement, U.P. (eds.) Non–Classical Logics and their Applications to Fuzzy Subsets, pp. 53–106. Kluwer Academic Publishers, Dordrecht (1996)Google Scholar
  11. 11.
    Jónsson, B., Tarski, A.: Boolean algebras with operators. Part I. American Journal of Mathematics 73, 891–939 (1951)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Koppelberg, S.: General Theory of Boolean algebras. North Holland, Amsteram (1989)Google Scholar
  13. 13.
    Lenzen, W.: Recent work in epistemic logic. Acta Philosophica Fennica 30, 1–129 (1978)MathSciNetGoogle Scholar
  14. 14.
    Meyer, J.J.C., van der Hoek, W.: Epistemic Logic for AI and Computer Science. Cambridge Tracks in Theoretical Computer Science, vol. 41. Cambridge University Press, Cambridge (1995)zbMATHCrossRefGoogle Scholar
  15. 15.
    Orłowska, E.: Representation of vague information. ICS PAS Report 503 (1983)Google Scholar
  16. 16.
    Orłowska, E.: Semantics of vague concepts. In: Dorn, G., Weingartner, P. (eds.) Foundations of Logic and Linguistics. Problems and Their Solutions. Selected Contributions to the 70th Intenational Congress of Logic Methodology and Philosophy of Science, Salzburg, 1983, pp. 465–482. Plenum Press, New York and London (1985)Google Scholar
  17. 17.
    Orłowska, E.: Logic for reasoning about knowledge. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 35, 556–568 (1989)Google Scholar
  18. 18.
    Orłowska, E., Radzikowska, A.M.: Relational Representability for Algebras of Substructural Logics. In: MacCaull, W., Winter, M., Düntsch, I. (eds.) RelMiCS 2005. LNCS, vol. 3929, pp. 212–226. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  19. 19.
    Orłowska, E., Radzikowska, A.M.: Representation theorems for some fuzzy logics based on residuated non–distributive lattices. Fuzzy Sets and Systems 159, 1247–1259 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Orłowska, E., Radzikowska, A.M.: Discrete duality for some axiomatic extensions of MTL algebras. In: Cintula, P., Hanikowá, Z., Svejdar, V. (eds.) Witnessed Years. Essays in Honour of Petr Hájek, pp. 329–344. College Publications, King’s College London (2010)Google Scholar
  21. 21.
    Orłowska, E., Rewitzky, I.: Duality via truth: Semantic frameworks for lattice–based logics. Logic Journal of the IGPL 13, 467–490 (2005)zbMATHCrossRefGoogle Scholar
  22. 22.
    Orłowska, E., Rewitzky, I.: Discrete Duality and Its Applications to Reasoning with Incomplete Information. In: Kryszkiewicz, M., Peters, J.F., Rybiński, H., Skowron, A. (eds.) RSEISP 2007. LNCS (LNAI), vol. 4585, pp. 51–56. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  23. 23.
    Orłowska, E., Rewitzky, I.: Context Algebras, Context Frames, and Their Discrete Duality. In: Peters, J.F., Skowron, A., Rybiński, H. (eds.) Transactions on Rough Sets IX. LNCS, vol. 5390, pp. 212–229. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  24. 24.
    Orłowska, E., Rewitzky, I.: Algebras for Galois–style connections and their discrete duality. Fuzzy Sets and Systems 161(9), 1325–1342 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Orłowska, E., Rewitzky, I., Düntsch, I.: Relational Semantics Through Duality. In: MacCaull, W., Winter, M., Düntsch, I. (eds.) RelMiCS 2005. LNCS, vol. 3929, pp. 17–32. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  26. 26.
    Pagliani, P.: Rough sets and Nelson algebras. Fundamenta Informaticae 27(2–3), 205–219 (1996)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Pagliani, P., Chakraborty, M.: A Geometry of Approximation. Trends in Logic, vol. 27. Springer (2008)Google Scholar
  28. 28.
    Parikh, R.: Knowledge and the problem of logical omniscience. In: Raś, W.Z., Zemankova, M. (eds.) Proceedings of the Second International Symposium on Methodologies for Intelligent Systems (ISMIS 1987), Charlotte, North Carolina, USA, October 14-17, pp. 432–439. North-Holland/Elsevier, Amsterdam (1987)Google Scholar
  29. 29.
    Parikh, R.: Logical omniscience. In: Leivant, D. (ed.) LCC 1994. LNCS, vol. 960, pp. 22–29. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  30. 30.
    Pawlak, Z.: Information systems – theoretical foundations. Information Systems 6, 205–218 (1981)zbMATHCrossRefGoogle Scholar
  31. 31.
    Pawlak, Z.: Rough sets. International Journal of Computer and Information Sciences 11(5), 341–356 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Pawlak, Z.: Rough sets - Theoretical Aspects of Reasoning about Data. Kluwer Academic Publishers, Dordrecht (1991)zbMATHGoogle Scholar
  33. 33.
    Radzikowska, A.M.: Discrete dualities for some information algebras based on De Morgan lattices (2011) (submitted)Google Scholar
  34. 34.
    Rasiowa, H., Sikorski, R.: Mathematics of Metamathematics. PWN, Warsaw (1963)zbMATHGoogle Scholar
  35. 35.
    Read, S.: Thinking about Logic. Oxford University Press, Oxford (1994)Google Scholar
  36. 36.
    Stone, M.: The theory of representations of Boolean algebras. Transactions of the American Mathematical Society 40(1), 37–111 (1936)MathSciNetGoogle Scholar
  37. 37.
    Wansing, H.: A general possible worlds framework for reasoning about knowledge and belief. Studia Logica 49(4), 523–539 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    von Wright, G.H.: An Essay in Modal Logic. North-Holland Publishing Company, Amsterdam (1952)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.National Institute of TelecommunicationsWarsawPoland
  2. 2.Faculty of Mathematics and Information ScienceWarsaw University of TechnologyWarsawPoland

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