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Logical and Algebraic Techniques for Rough Set Data Analysis

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Rough Set Methods and Applications

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 56))

Abstract

Abstract. In this paper, we shall give an introduction to, and an overview of, the various relational, algebraic, and logical tools that are available to handle rule based reasoning.

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References

  1. Balbes, R. and Dwinger, P. (1974). Distributive Lattices. University of Missouri Press, Columbia.

    MATH  Google Scholar 

  2. Balbiani, P. (1997). Axiomatisation of logics based on Kripke models with relative accessibility relations. In [55].

    Google Scholar 

  3. Balbiani, P. and Orlowska, E. (1999). A hierarchy of modal logics with relative accessibility relations. Journal of Applied Non—Classical Logics. To appear.

    Google Scholar 

  4. Becchio, D. (1978). Logique trivalente de Lukasiewicz. Ann. Sci. Univ. Clermont-Ferrand, 16:38–89.

    Google Scholar 

  5. Bennett, B. (1998). Modal semantics for knowledge bases dealing with vague concepts. Submitted for publication.

    Google Scholar 

  6. Birkhoff, G. (1948). Lattice Theory, volume 25 of Am. Math. Soc. Colloquium Publications. AMS, Providence, 2 edition.

    Google Scholar 

  7. Boicescu, V., Filipoiu, A., Georgescu, G., and Rudeanu, S. (1991). Lukasiewicz— Moisil Algebras, volume 49 of Annals of Discrete Mathematics. North Holland, Amsterdam.

    Google Scholar 

  8. Bull, R. and Segerberg, K. (1984). Basic modal logic. In [33], pages 1–88.

    Google Scholar 

  9. Comer, S. (1991). An algebraic approach to the approximation of information. Fundamenta Informaticae, 14:492–502.

    MathSciNet  MATH  Google Scholar 

  10. Comer, S. (1993). On connections between information systems, rough sets, and algebraic logic. In Rauszer, C., editor, Algebraic Methods in Logic and Cornputer Science, volume 28 of Banach Center Publications, pages 117–124. Polish Academy of Science, Warszawa.

    Google Scholar 

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

    MATH  Google Scholar 

  12. Demri, S. (1996). A class of information logics with a decidable validity problem. In 21st International Symposium on Mathematical Foundations of Computer Science, volume 1113 of LNCS, pages 291–302. Springer-Verlag.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Demri, S. and Orlowska, E. (1998). Informational representability of models for information logic. In Orlowska, E., editor, Logic at Work. Physica — Verlag, Heidelberg. To appear.

    Google Scholar 

  15. Dntsch, I. (1983). On free or projective Stone algebras. Houston J. Math., 9:455–463.

    MathSciNet  Google Scholar 

  16. Düntsch, I. (1994). Rough relation algebras. Fundamenta Informaticae, 21:321–331.

    MathSciNet  MATH  Google Scholar 

  17. Diintsch, I. (1997). A logic for rough sets. Theoretical Computer Science, 179(12):427–436.

    Article  MathSciNet  Google Scholar 

  18. Diintsch, I. and Gediga, G. (1996). On query procedures to build knowledge structures. J. Math. Psych., 40(2):160–168.

    Article  MathSciNet  Google Scholar 

  19. Düntsch, I. and Gediga, G. (1997). Algebraic aspects of attribute dependencies in information systems. Fundamenta Informaticae, 29:119–133.

    MathSciNet  MATH  Google Scholar 

  20. Düntsch, I. and Gediga, G. (1998). Simple data filtering in rough set systems. International Journal of Approximate Reasoning, 18(1–2):93–106.

    Article  MathSciNet  MATH  Google Scholar 

  21. Düntsch, I. and Gediga, G. (1999). A note on the correspondences among entail relations, rough set dependencies, and logical consequence. Submitted for publication.

    Google Scholar 

  22. Düntsch, I., Gediga, G., and Orlowska, E. (1999a). Relational attribute systems. Submitted for publication.

    Google Scholar 

  23. Düntsch, I. and Orlowska, E. (1999). Mixing modal and sufficiency operators. Bulletin of the Section of Logic, Polish Academy of Sciences, 28(2):99–106.

    MATH  Google Scholar 

  24. Düntsch, I. and Orlowska, E. (2000a). Beyond modalities: Sufficiency and mixed algebras. Algebra Universalis. Submitted for publication.

    Google Scholar 

  25. Diintsch, I. and Orlowska, E. (2000b). Logics of complementarity in information systems. Mathematical Logic Quarterly, 46(4). To appear.

    Google Scholar 

  26. Dntsch, I. and Orlowska, E. (2000c). A proof system for contact relation algebras. Journal of Philosophical Logic. To appear.

    Google Scholar 

  27. Düntsch, I., Orlowska, E., and Wang, H. (2000a). An algebraic and logical approach to the approximation of regions. In Proc of the 5th Seminar of Relational Methods in Computer Science, Banff, Jan 2000. To appear.

    Google Scholar 

  28. Düntsch, I., Orlowska, E., and Wang, H. (2000b). Approximating regions. In preparation.

    Google Scholar 

  29. Düntsch, I., Wang, H., and McCloskey, S. (1999b). Relation algebras in qualitative spatial reasoning. Fundamenta Informaticae, 39:229–248.

    MathSciNet  Google Scholar 

  30. Falmagne, J.-C., Koppen, M., Villano, M., Doignon, J.-P., and Johannesen, J. (1990). Introduction to knowledge spaces: How to build, test and search them. Psychological Review, 97:202–234.

    Article  Google Scholar 

  31. Fitting, M. (1993). Basic modal logic. In Gabbay, D. M., Hogger, C. J., and Robinson, J. A., editors, Logical foundations, volume 1 of Handbook of Logic in Artificial Intelligence and Logic Programming, pages 368–448. Clarendon Press, Oxford.

    Google Scholar 

  32. Gabbay, D. M. (1985). Theoretical foundations for non-monotonic reasoning in expert systems. In Apt, K. R., editor, Logics and Models of Concurrent Systems, volume F13 of NATO Advanced Studies Institute, pages 439–457. Springer, Berlin.

    Google Scholar 

  33. Gabbay, D. M. and Guenthner, F., editors (1984). Extensions of classical logic, volume 2 of Handbook of Philosophical Logic. Reidel, Dordrecht.

    Google Scholar 

  34. Gargov, G., Passy, S., and Tinchev, T. (1987). Modal environment for Boolean speculations. In Skordev, D., editor, Mathematical Logic and Applications, pages 253–263, New York. Plenum Press.

    Chapter  Google Scholar 

  35. Gediga, G. and Düntsch, I. (2000). Statistical techniques for rough set data analysis. In Polkowski, L., editor, Rough sets: New developments. Physica Verlag, Heidelberg. To appear.

    Google Scholar 

  36. Düntsch, G. (1978). General Lattice Theory. Birkhäuser, Basel.

    Google Scholar 

  37. Humberstone, I. L. (1983). Inaccessible worlds. Notre Dame Journal of Formal Logic, 24:346–352.

    Article  MathSciNet  MATH  Google Scholar 

  38. Iturrioz, L. and Orlowska, E. (1996). A Kripke-style and relational semantics for logics based on Lukasiewicz algebras. Conference in honour of J. Lukasiewicz, Dublin.

    Google Scholar 

  39. Iwinski, T. B. (1987). Algebraic approach to rough sets. Bull. Polish Acad. Sci. Math., 35:673–683.

    MathSciNet  MATH  Google Scholar 

  40. Jónsson, B. and Tarski, A. (1951). Boolean algebras with operators I. Amer. J. Math., 73:891–939.

    Article  MathSciNet  MATH  Google Scholar 

  41. Jónsson, B. and Tarski, A. (1952). Boolean algebras with operators II. Amer. J. Math., 74:127–162.

    Article  MATH  Google Scholar 

  42. Konikowska, B. (1987). A formal language for reasoning about indiscernibility. Bulletin of the Polish Academy of Sciences, Mathematics, 35:239–249.

    MathSciNet  MATH  Google Scholar 

  43. Konikowska, B. (1997). A logic for reasoning about similarity. Studia Logica, 58:185–226.

    Article  MathSciNet  MATH  Google Scholar 

  44. Koppen, M. and Doignon, J.-P. (1990). How to build a knowledge space by querying an expert. J. Math. Psych., 34:311–331.

    Article  MathSciNet  MATH  Google Scholar 

  45. Novotný, M. (1997a). Applications of dependence spaces. In [55], pages 247–289.

    Google Scholar 

  46. Novotný, M. (1997b). Dependence spaces of information systems. In [55], pages 193–246.

    Google Scholar 

  47. Orlowska, E. (1984). Modal logics in the theory of information systems. Zeitschr. f. Math. Logik und Grundlagen der Math., 30:213–222.

    Article  MathSciNet  MATH  Google Scholar 

  48. Orlowska, E. (1988a). Kripke models with relative accessibility relations and their applications to inference with incomplete information. In Mirkowska, G. and Rasiowa, H., editors, Mathematical Problems in Computation Theory, volume 21 of Banach Center Publications, pages 327–337. PWN.

    Google Scholar 

  49. Orlowska, E. (1988b) Logical aspects of learning concepts. Journal of Approximate Reasoning, 2:349–364.

    Article  MATH  Google Scholar 

  50. Orlowska, E. (1989). Logic for reasoning about knowledge. Zeitschr. f. Math. Logik und Grundlagen der Math., 35:559–572.

    Article  MathSciNet  MATH  Google Scholar 

  51. Orlowska, E. (1990). Kripke semantics for knowledge representation logics. Studia Logica, 49:255–272.

    Article  MathSciNet  MATH  Google Scholar 

  52. Orlowska, E. (1991). Relational interpretation of modal logics. In Andréka, H., Monk, J. D., and Németi, I., editors, Algebraic Logic, volume 54 of Colloquia Mathematica Societatis János Bolyai, pages 443–471. North Holland, Amsterdam.

    Google Scholar 

  53. Orlowska, E. (1995). Information algebras. In Proceedings of AMAST 95, volume 639 of Lecture Notes in Computer Science. Springer-Verlag.

    Google Scholar 

  54. Orlowska, E. (1996). Relational proof systems for modal logics. In Wansing, H., editor, Proof theory of modal logic, pages 55–78. Kluwer, Dordrecht.

    Google Scholar 

  55. Orlowska, E., editor (1997a). Incomplete Information — Rough Set Analysis. Physica — Verlag, Heidelberg.

    Google Scholar 

  56. Orlowska, E. (1997b). Introduction: What you always wanted to know about rough sets. In [55], pages 1–20.

    Google Scholar 

  57. Orlowska, E. and Pawlak, Z. (1987). Representation of nondeterministic information. Theoretical Computer Science, 29:27–39.

    Article  MathSciNet  Google Scholar 

  58. Pagliani, P. (1997). Rough sets theory and logic-algebraic structures. In [55], pages 109–190.

    Google Scholar 

  59. Pawlak, Z. (1973). Mathematical foundations of information retrieval. ICS Research Report 101, Polish Academy of Sciences.

    Google Scholar 

  60. Pawlak, Z. (1991). Rough sets: Theoretical aspects of reasoning about data, volume 9 of System Theory, Knowledge Engineering and Problem Solving. Kluwer, Dordrecht.

    Google Scholar 

  61. Pomykala, J. and Pomykala, J. A. (1988). The Stone algebra of rough sets. Bull. Polish Acad. Sci. Math., 36:495–508.

    MathSciNet  MATH  Google Scholar 

  62. Rasiowa, H. and Sikorski, R. (1963). The Mathematics of Metamathematics, volume 41 of Polska Akademia Nauk. Monografie matematyczne. PWN, Warsaw.

    Google Scholar 

  63. Tarski, A. (1930). Fundamentale Begriffe der Methodologie der deduktiven Wissenschaften. Monatsh. Math. Phys., 37:361–404.

    Article  MathSciNet  MATH  Google Scholar 

  64. Vakarelov, D. (1977). Notes on N-lattices and constructive logic with strong negation. Studia Logica, 36:109–125.

    Article  MathSciNet  MATH  Google Scholar 

  65. Vakarelov, D. (1989). Modal logics for knowledge representation systems. In Meyer, A. R. and Zalessky, M., editors, Symposium on Logic Foundations of Computer Science, Pereslavl-Zalessky, volume 363 of LNCS, pages 257–277. Springer-Verlag.

    Google Scholar 

  66. Vakarelov, D. (1991). Modal logics for knowledge representation systems. Theoretical Computer Science, 90:433–456.

    MathSciNet  MATH  Google Scholar 

  67. Vakarelov, D. (1992). Consequence relations and information systems. In Slowiński, R., editor, Intelligent decision support: Handbook of applications and advances of rough set theory, volume 11 of System Theory, Knowledge Engineering and Problem Solving, pages 391–399. Kluwer, Dordrecht.

    Google Scholar 

  68. Vakarelov, D. (1997a). Information systems, similarity relations, and modal logics. In 155], pages 492–550.

    Google Scholar 

  69. Vakarelov, D. (1997b). Modal logics of arrows. In de Rijke, M., editor, Advances in intensional logic, pages 137–171. Kluwer.

    Google Scholar 

  70. van Benthem, J. (1984). Correspondence theory. In [33], pages 167–247.

    Google Scholar 

  71. Varlet, J. C. (1968). Algèbres des Lukasiewicz trivalentes. Bull. Soc. Roy. Sci. Liège, 36:399–408.

    MathSciNet  Google Scholar 

  72. Varlet, J. C. (1972). A regular variety of type (2,2,1,1,0,0). Algebra Universalis, 2:218–223.

    Article  MathSciNet  MATH  Google Scholar 

  73. Wang, H., Düntsch, I., and Gediga, G. (2000). Classificatory filtering in decision systems. International Journal of Approximate Reasoning. To appear.

    Google Scholar 

  74. Worboys, M. (1998). Imprecision in finite resolution spatial data. Geoinformatica, 2(3):257–280.

    Article  Google Scholar 

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Düntsch, I., Gediga, G. (2000). Logical and Algebraic Techniques for Rough Set Data Analysis. In: Polkowski, L., Tsumoto, S., Lin, T.Y. (eds) Rough Set Methods and Applications. Studies in Fuzziness and Soft Computing, vol 56. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-1840-6_10

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  • DOI: https://doi.org/10.1007/978-3-7908-1840-6_10

  • Publisher Name: Physica, Heidelberg

  • Print ISBN: 978-3-662-00376-3

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