Advertisement

Rough Sets pp 361-410 | Cite as

Algebra and Logic of Rough Sets

  • Lech Polkowski
Part of the Advances in Soft Computing book series (AINSC, volume 15)

Abstract

In this Chapter, we discuss algebraic structures induced in collections of rough sets and we present two logical structures, rooted respectively in intuitionistic and modal logics, which reflect properties of indiscernibility and tolerance relations that arise in the attribute¡ªvalue formalization of information systems. The foundations for this discussion were laid in the papers [Pawlak 81b, 82b, 87], (Orlowska—Pawlak 84a,b].

Keywords

Modal Logic Heyting Algebra Axiom Scheme Information Logic Stone Algebra 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Works quoted

  1. [Archangelsky—Taitslin97]
    D. A. Archangelsky and M. A. Taitslin, A logic for information systems, Studia Logica, 58 (1997), pp. 3–16.MathSciNetMATHCrossRefGoogle Scholar
  2. [Banerjee—Chakraborty98]
    M. Banerjee and M. K. Chakraborty, Rough logics: a survey with further directions, in: E. Orlowska (ed.), Incomplete Information: Rough Set Analysis, Studies in Fuzziness and Soft Computing, vol. 13, Physica Verlag, Heidelberg, 1998, pp. 579–600.Google Scholar
  3. [Becchio78]
    D. Becchio, Logique trivalente de Lukasiewicz, Ann. Sci. Univ. Clermont—Ferrand, 16 (1978), pp. 38–89.Google Scholar
  4. [Becchio72]
    D. Becchio, Nouvelle démonstration de la complétude du systéme de Wajsberg axiomatisant la logique trivalente de Lukasiewicz, C. R. Paris, 275 (1972), pp. 679–681.MathSciNetMATHGoogle Scholar
  5. [Boicescu9l]
    V. Boicescu, A. Filipoiu, S. Georgescu and S. Rudeanu, Lukasiewicz—Moisil Algebras, North Holland, Ansterdam, 1991.MATHGoogle Scholar
  6. [Cattaneo98]
    G. Cattaneo, Abstract approximation spaces for rough theories,in: [Orlowska98], pp. 59–98.Google Scholar
  7. [Obtulowicz85]
    A. Obtulowicz, Rough sets and Heyting algebra valued sets, Bull. Polish Acad. Sci. Math., 33 (1985), pp. 454–476.Google Scholar
  8. [Cattaneo97]
    G. Cattaneo, Generalized rough sets. Preclusivity fuzzy—intuitionistic (BZ) lattices, Studia Logica, 58 (1997), pp. 47–77.MathSciNetMATHCrossRefGoogle Scholar
  9. [Cignoli69]
    R. Cignoli, Algebras de Moisil de orden n, Doctoral Thesis, Univ. Nacional del Sur, Bahia Blanca, Brasil, 1969.Google Scholar
  10. [Comer93]
    S. Comer, On connections between information systems, rough sets and algebraic logic, in: Algebraic Methods in Logic and Computer Science, Banach Center Publ., 28, Warszawa, 1993.Google Scholar
  11. [Duentsch98]
    I. Duentsch, Rough sets and algebras of relations,in: [Orlowska98], pp. 95–108.Google Scholar
  12. [Duentsch94]
    I. Duentsch, Rough relation algebras, Fundamenta Informaticae, 21 (1994), pp. 321–331.Google Scholar
  13. [Epstein60]
    G. Epstein, The lattice theory of Post algebras, Trans. Amer. Math. Soc., 95 (1960), 300–317.Google Scholar
  14. [Gentzen34]
    G. Gentzen, Untersuchungen über das logische Schliessen. I, II, Mathematische Zeitschrift 39 (1934–5), pp. 176–210, 405–431.Google Scholar
  15. [Iturrioz77]
    L. Iturrioz, Lukasiewicz and symmetrical Heyting algebras, Zeit. Math. Logik u. Grundl. Math., 23 (1977), pp. 131–136.Google Scholar
  16. [Iwinski88]
    T. B. Iwinski, Rough orders and rough set concepts, Bull. Polish Acad. Ser. Sci. Math., 37 (1988), pp. 187–192.Google Scholar
  17. [Iwinski87]
    T. B. Iwinski, Algebraic approach to rough sets, Bull. Polish Acad. Ser. Sci. Math., 35 (1987), pp. 673–683.Google Scholar
  18. [Moisi164]
    Gr. C. Moisil, Sur les logiques de Lukasiewicz d un nombre fini de valeurs, Rev. Roumaine Math. Pures Appl., 9 (1964), pp. 905–920, 583–595.Google Scholar
  19. [Moisil63]
    Gr. C. Moisil, Les logiques non—chrysippiennes et leurs applications, Acta Phil. Fennica, 16 (1963), pp. 137–152.Google Scholar
  20. [Moisi160]
    Gr. C. Moisil, Sur les idéaux des algébres lukasiewicziennes trivalentes, An. Univ. C. I. Parhon, Acta logica, 3 (1960), pp. 83–95, 244–258.Google Scholar
  21. [Moisi142]
    Gr. C. Moisil, Logique modale, Disquisitiones Math. Phys., 2 (1942), pp. 3–98, 217–328, 341–441.Google Scholar
  22. [Post21]
    E. Post, Introduction to a general theory of elementary propositions, Amer. J. Math., 43 (1921), pp. 163–185.Google Scholar
  23. [Nakamura98]
    A. Nakamura, Graded modalities in rough logic, in: L. Polkowski and A. Skowron (eds.), Rough Sets in Knowledge Discovery. Methodology and Applications, Studies in Fuzziness and Soft Computing, vol. 18, Physica Verlag, Heidelberg, 1998, pp. 192–208.Google Scholar
  24. [Nelson49]
    D. Nelson, Constructible falsity, The Journal of Symbolic Logic, 14 (1949), pp. 16–26.Google Scholar
  25. [Orlowska98]
    E. Orlowska, ed., Incomplete Information: Rough Set Analysis, Studies in Fuzzines and Soft Computing vol. 13, Physica Verlag, Heidelberg, 1998.Google Scholar
  26. [Orlowska89]
    E. Orlowska, Logic for reasoning about knowledge, Z. Math. Logik u. Grund. d. Math., 35 (1989), pp. 559–572.MathSciNetMATHCrossRefGoogle Scholar
  27. [Orlowska85]
    E. Orlowska, Logic approach to information systems, Fundamenta Informaticae, 8 (1985), pp. 359–378.MathSciNetGoogle Scholar
  28. [Orlowska84]
    E. Orlowska, Modal logics in the theory of info nation systems,Z. Math. Logik u. Grund.d. Math., 30 (1984), pp. 213–222.Google Scholar
  29. [Orlowska-Pawlak84a]
    E. Orlowska and Z. Pawlak, Logical foundations of knowledge representation, Reports of the Comp. Centre of the Polish Academy of Sciences, 537, 1984.Google Scholar
  30. [Orlowska-Pawlak84b]
    E. Orlowska and Z. Pawlak, Representation of non-deterministic information, Theor. Computer Science, 29 (1984), pp. 27–39.MathSciNetGoogle Scholar
  31. [Pagliani98a]
    P. Pagliani, Rough set theory and logic-algebraic structures,in: [Orlowska98], pp. 109–192.Google Scholar
  32. [Pagliani98b]
    P. Pagliani, A practical introduction to the modal-relational approach to approximation spaces,in: [Polkowski-Skowron98a], pp. 209–232.Google Scholar
  33. [Pagliani96]
    P. Pagliani, Rough sets and Nelson algebras, Fundamenta informaticae, 27 (1996), pp. 205–219.MathSciNetGoogle Scholar
  34. [Pawlak87b]
    Z. Pawlak, Rough logic, Bull. Polish Acad. Sci. Tech., 35 (1987), pp. 253–258.Google Scholar
  35. [Pawlak82b]
    Z. Pawlak, Rough sets, algebraic and topological approach, Int. J. Inform. Comp. Sciences, 11 (1982), pp. 341–366.Google Scholar
  36. [Pawlak8lb]
    Z. Pawlak, Information systems-theoretical foundations, Information Systems, 6 (1981), pp. 205–218.Google Scholar
  37. [Pomykala88]
    J. Pomykala and J. A. Pomykala, The Stone algebra of rough sets, Bull. Polish Acad. Ser. Sci. Math., 36 (1988), 495–508.MathSciNetMATHGoogle Scholar
  38. [Rasiowa74]
    H. Rasiowa, An Algebraic Approach to Non-Classical Logics, PWN-Polish Scientific Publishers — North-Holland, Warszawa—Amsterdam, 1974.MATHGoogle Scholar
  39. [Rasiowa—Skowron86a]
    H. Rasiowa and A. Skowron, Rough concept logic, LNCS vol. 208, Springer Verlag, Berlin, 1986, pp. 288–297.Google Scholar
  40. [Rasiowa—Skowron86b]
    H. Rasiowa and A. Skowron, The first step towards an approximation logic, J. Symbolic Logic, 51 (1986), p. 509.Google Scholar
  41. [Rasiowa—Skowron86c]
    H. Rasiowa and A. Skowron, Approximation logic, Proc. Conf. on Mathematical Methods of Specification and Synthesis of Software Systems, Akademie Verlag, Berlin, 1986, pp. 123–139.Google Scholar
  42. [Rasiowa—Skowron84]
    H. Rasiowa and A. Skowron, A rough concept logic, in: A. Skowron (ed.), Proc. the 5th Symposium on Comp. Theory, Lecture Notes in Computer Science, vol. 208 (1984), pp. 197–227.Google Scholar
  43. [Rauszer85a]
    C. M. Rauszer, Dependency of attributes in information systems, Bull. Polish Acad. Sci. Math., 33 (1985), pp. 551–559.Google Scholar
  44. [Rauszer85b]
    C. M. Rauszer, An equivalence between theory of functional dependencies and a fragment of intuitionistic logic, Bull. Polish Acad. Sci. Math., 33 (1985), pp. 571–579.Google Scholar
  45. [Rauszer84]
    C. M. Rauszer, An equivalence between indiscernibility relations in information systems and a fragment of intuitionistic logic, in: A. Skowron (ed.), Proc. the 5th Symp. Comp. Theory, Lecture Notes in Computer Science, vol. 208, Springer Verlag, Berlin, 1984, pp. 298–317.Google Scholar
  46. [Rosenbloom42]
    P. Rosenbloom, Post algebras.I. Postulates and general theory, Amer. J. Math., 64 (1942), pp. 167–183.Google Scholar
  47. [Rousseau70]
    G. Rousseu, Post algebras and pseudo—Post algebras, Fund. Math., 67 (1970), pp. 133–145.Google Scholar
  48. [Traczyk63]
    T. Traczyk, Axioms and some properties of Post algebras, Collor. Math., 10 (1963), pp. 193–209.Google Scholar
  49. [Vakarelov89]
    D. Vakarelov, Modal logics for knowledge representation systems, Lecture Notes in Computer Science, vol. 363 (1989), pp. 257–277.Google Scholar
  50. [Varlet68]
    J. Varlet, Algébres de Lukasiewicz trivalentes, Bull. Soc. Roy.Google Scholar
  51. [Wajsberg3l]
    M. Wajsberg, Axiomatization of the three—valued sentential calculus (in Polish, Summary in German), C. R. Soc. Sci. Lettr. Varsovie, 24 (1931), 126–148.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Lech Polkowski
    • 1
    • 2
  1. 1.Polish-Japanese Institute of Information TechnologyWarsawPoland
  2. 2.Department of Mathematics and Computer ScienceUniversity of Wormia and MazuryOlsztynPoland

Personalised recommendations