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Knowledge Algebras and Their Discrete Duality

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Rough Sets and Intelligent Systems - Professor Zdzisław Pawlak in Memoriam

Part of the book series: Intelligent Systems Reference Library ((ISRL,volume 43))

Abstract

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.

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References

  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)

    Chapter  Google Scholar 

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

    Book  MATH  Google Scholar 

  3. Davey, B.A., Priestley, H.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  4. Demri, S.: A completeness proof for a logic with an alternative necessity operator. Studia Logica 58, 99–112 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Demri, S.: A logic with relative knowledge operators. Journal of Logic, Language and Information 8, 167–185 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Demri, S., Orłowska, E.: Incomplete Information: Structure, Inference, Complexity. EATCS Monographs in Teoretical Computer Science. Springer, Heidelberg (2002)

    MATH  Google Scholar 

  7. Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning about Knowledge. MIT Press, Cambridge (1995)

    MATH  Google Scholar 

  8. Hintikka, J.: Knowledge and Belief. Cornell University Press, London (1962)

    Google Scholar 

  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. 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. Jónsson, B., Tarski, A.: Boolean algebras with operators. Part I. American Journal of Mathematics 73, 891–939 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  12. Koppelberg, S.: General Theory of Boolean algebras. North Holland, Amsteram (1989)

    Google Scholar 

  13. Lenzen, W.: Recent work in epistemic logic. Acta Philosophica Fennica 30, 1–129 (1978)

    MathSciNet  Google Scholar 

  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)

    Book  MATH  Google Scholar 

  15. Orłowska, E.: Representation of vague information. ICS PAS Report 503 (1983)

    Google Scholar 

  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. Orłowska, E.: Logic for reasoning about knowledge. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 35, 556–568 (1989)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Orłowska, E., Rewitzky, I.: Duality via truth: Semantic frameworks for lattice–based logics. Logic Journal of the IGPL 13, 467–490 (2005)

    Article  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  24. Orłowska, E., Rewitzky, I.: Algebras for Galois–style connections and their discrete duality. Fuzzy Sets and Systems 161(9), 1325–1342 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  26. Pagliani, P.: Rough sets and Nelson algebras. Fundamenta Informaticae 27(2–3), 205–219 (1996)

    MathSciNet  MATH  Google Scholar 

  27. Pagliani, P., Chakraborty, M.: A Geometry of Approximation. Trends in Logic, vol. 27. Springer (2008)

    Google Scholar 

  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. Parikh, R.: Logical omniscience. In: Leivant, D. (ed.) LCC 1994. LNCS, vol. 960, pp. 22–29. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  30. Pawlak, Z.: Information systems – theoretical foundations. Information Systems 6, 205–218 (1981)

    Article  MATH  Google Scholar 

  31. Pawlak, Z.: Rough sets. International Journal of Computer and Information Sciences 11(5), 341–356 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  32. Pawlak, Z.: Rough sets - Theoretical Aspects of Reasoning about Data. Kluwer Academic Publishers, Dordrecht (1991)

    MATH  Google Scholar 

  33. Radzikowska, A.M.: Discrete dualities for some information algebras based on De Morgan lattices (2011) (submitted)

    Google Scholar 

  34. Rasiowa, H., Sikorski, R.: Mathematics of Metamathematics. PWN, Warsaw (1963)

    MATH  Google Scholar 

  35. Read, S.: Thinking about Logic. Oxford University Press, Oxford (1994)

    Google Scholar 

  36. Stone, M.: The theory of representations of Boolean algebras. Transactions of the American Mathematical Society 40(1), 37–111 (1936)

    MathSciNet  Google Scholar 

  37. Wansing, H.: A general possible worlds framework for reasoning about knowledge and belief. Studia Logica 49(4), 523–539 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  38. von Wright, G.H.: An Essay in Modal Logic. North-Holland Publishing Company, Amsterdam (1952)

    Google Scholar 

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Author information

Authors and Affiliations

  1. National Institute of Telecommunications, ul. Szachowa 1, Warsaw, Poland

    Ewa Orłowska

  2. Faculty of Mathematics and Information Science, Warsaw University of Technology, Plac Politechniki 1, 00-661, Warsaw, Poland

    Anna Maria Radzikowska

Authors
  1. Ewa Orłowska
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  2. Anna Maria Radzikowska
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Corresponding author

Correspondence to Ewa Orłowska .

Editor information

Editors and Affiliations

  1. Institute of Mathematics, Warsaw University, Banacha 2, Warsaw, 02097, Poland

    Andrzej Skowron

  2. Institute of Computer Science, University of Rzeszów, ul. Dekerta 2, Rzeszów, 35-030, Poland

    Zbigniew Suraj

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Orłowska, E., Radzikowska, A.M. (2013). Knowledge Algebras and Their Discrete Duality. In: Skowron, A., Suraj, Z. (eds) Rough Sets and Intelligent Systems - Professor Zdzisław Pawlak in Memoriam. Intelligent Systems Reference Library, vol 43. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30341-8_2

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  • DOI: https://doi.org/10.1007/978-3-642-30341-8_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-30340-1

  • Online ISBN: 978-3-642-30341-8

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