Skip to main content
SpringerLink
Log in

Rough Sets and Rough Logic: A KDD Perspective

  • Chapter
Rough Set Methods and Applications

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

Abstract

Basic ideas of rough set theory were proposed by Zdzislaw Pawlak [90, 91] in the early 1980’s. In the ensuing years, we have witnessed a systematic, world-wide growth of interest in rough sets and their applications. There are numerous areas of successful applications of rough set software systems [101]. Many interesting case studies are reported (for references see e.g., [100, 101], [87] and the bibliography in these books, in particular [19], [46], [57], [132], [1461).

The main goal of rough set analysis is induction of approximations of concepts. This main goal is motivated by the basic fact, constituting also the main problem of KDD, that languages we may choose for knowledge description are incomplete with respect to expressibility. A fortiori, we have to describe concepts of interest (features, properties, relations etc.) known not completely but by means of their reflections (i.e., approximations) in the chosen language. The most important issues in this induction process are:

  • construction of relevant primitive concepts from which approximations of more complex concepts are assembled,

  • measures of inclusion and similarity (closeness) on concepts,

  • construction of operations producing complex concepts from the primitive ones.

Basic tools of rough set approach are related to concept approximations. They are defined by approximation spaces. For many applications, in particular for KDD problems, it is necessary to search for relevant approximation spaces in the large space of parameterized approximation spaces. Strategies for tuning parameters of approximation spaces are crucial for inducing concept approximations of high quality.

Methods proposed in rough set approach are kin to general methods used to solve Knowledge Discovery and Data Mining (KDD) problems like feature selection, feature extraction (e.g., discretization or grouping of symbolic value), data reduction, decision rule generation, pattern extraction (templates, association rules), or decomposition of large data tables. In this Chapter we examine rough set contributions to Knowledge Discovery from the perspective of KDD as a whole.

This Chapter shows how several aspects of the above problems are solved by the classical rough set approach and how they are approached by some recent extensions to the classical theory of rough sets. We point out the role of Boolean reasoning in solving discussed problems. Rough sets induce via its methods a specific logic, which we call rough logic. We also discuss rough logic and related logics from a wider perspective of logical approach in KDD. We show some relationships between these logics and potential directions for further research on rough logic.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adams, E.W.: The Logic of Conditionals, An Application of Probability to Deductive Logic. D. Reidel Publishing Company, Dordrecht, 1975.

    MATH  Google Scholar 

  2. Agotnes, T., Komorowski, J., Loken, T.: Taming large rule models in rough set approaches. Proceedings of the 3rd European Conference of Principles and Practice of Knowledge Discovery in Databases, September 15–18, 1999, Prague, Czech Republic, Lecture Notes in Artificial Intelligence 1704, Springer-Verlag, Berlin, 1999, pp. 193–203.

    Google Scholar 

  3. Agrawal, R., Imieliński, T., Swami, A.: Mining association rules between sets of items in large databases. In: Proceedings ACM SIGMOD Conference on Management of Data, Washington, 1993, pp. 207–213.

    Google Scholar 

  4. Agrawal, R., Mannila, H., Srikant, R., Toivonen, H., Verkano, A.: Fast discovery of association rules. Fayyad UM., Piatetsky-Shapiro G., Smyth P., Uthurusamy R. (Eds.): Advances in Knowledge Discovery and Data Mining, The AAAI Press/The MIT Press 1996, pp. 307–328.

    Google Scholar 

  5. Anderson, A.R., Belnap. N.D.: Entailment: The Logic of Relevance and Necessity. Princeton University Press, Vol.1, 1975.

    MATH  Google Scholar 

  6. Anderson, A.R., Belnap, N.D., Dunn, J.M.: Entailment: Vol.2. Oxford University Press, 1992.

    MATH  Google Scholar 

  7. Anthony, M., Biggs, N.: Computational Learning Theory. Cambridge University Press, Cambridge, 1992.

    MATH  Google Scholar 

  8. Aquist, L.: Deontic logic. In: Gabbay, D., Guenthner, F. (eds.): Handbook of Philosophical Logic Vol.2. Kluwer Academic Publishers, Dordrecht, 1994, pp. 605–714.

    Google Scholar 

  9. Bazan, J.G., Nguyen, Son Hung, Nguyen, Tuan Trung, Skowron, A., Stepaniuk, J.: Decision rules synthesis for object classification. In: E. Orlowska (ed.), Incomplete Information: Rough Set Analysis, Physica — Verlag, Heidelberg, 1998, pp. 23–57.

    Google Scholar 

  10. Bazan, J.G.: A comparison of dynamic and non-dynamic rough set methods for extracting laws from decision system. In: Polkowski, L., Skowron, A. (Eds.): Rough Sets in Knowledge Discovery 1: Methodology and Applications. Physica-Verlag, Heidelberg, 1998, pp. 321–365.

    Google Scholar 

  11. van Bentham, J.: Temporal logic. In: Gabbay, D.M., Hogger, C.J., and Robinson, J.A. (eds.): Handbook of Logic in Artificial Intelligence and Logic Programming Vol.4. Oxford University Press, New York,1995, pp. 241–350.

    Google Scholar 

  12. van Bentham, J.: Logic after the Golden Age. IILC Magazine, December 1999, Institute for Logic, Language and Computation, Amsterdam University, 1999, p. 12.

    Google Scholar 

  13. Boole, G.: An Investigation of the Laws of Thought. Walton and Maberley, London, 1854.

    Google Scholar 

  14. Brown, EM.: Boolean Reasoning. Kluwer Academic Publishers, Dordrecht, 1990.

    MATH  Google Scholar 

  15. Cattaneo, G.: Abstract approximation spaces for rough theories. In: Polkowski, L., Skowron, A. (Eds.): Rough Sets in Knowledge Discovery 1: Methodology and Applications. Physica-Verlag, Heidelberg, 1998, pp. 59–98.

    Google Scholar 

  16. Chmielewski, M.R., Grzymala-Busse, J.W.: Global discretization of attributes as preprocessing for machine learning. In: Proceedings of the Third International Workshop on Rough Sets and Soft Computing (RSSC’94),San Jose State University, San Jose, California, USA, November 10–12, 1994, pp. 294–301.

    Google Scholar 

  17. Cios, J., Pedrycz, W., Swiniarski, R.W.: Data Mining in Knowledge Discovery. Kluwer Academic Publishers, Dordrecht, 1998.

    Google Scholar 

  18. Codd, E.F.: A relational model for large shared data banks. Comm. ACM. Vol.13, No 6, 1970, pp. 377–382.

    MATH  Google Scholar 

  19. Czyżewski, A.: Soft processing of audio signals. In: Polkowski, L., Skowron, A. (Eds.): Rough Sets in Knowledge Discovery 2: Applications, Case Studies and Software Systems. Physica-Verlag, Heidelberg, 1998, pp. 147–165.

    Google Scholar 

  20. Davis, M.: Computability and Unsolvability. Mc Graw-Hill, New York, 1958.

    MATH  Google Scholar 

  21. Deogun, J., Raghavan, V., Sarkar, A., Sever, H.: Data mining: Trends in research and development. In: Lin, T.Y., Cercone, N. (Eds.): Rough Sets and Data Mining. Analysis of Imprecise Data. Kluwer Academic Publishers, Boston, 1997, pp. 9–45.

    Google Scholar 

  22. Dougherty J., Kohavi R., and Sahami M.: Supervised and unsupervised discretization of continuous features. In: Proceedings of the Twelfth International Conference on Machine Learning, Morgan Kaufmann, San Francisco, CA, 1995.

    Google Scholar 

  23. Dubois, D., Prade, H.: Belief Change. Vol. 3 in: Gabbay, D.M., Smets Ph. (eds.), Handbook of Defeasible Reasoning and Uncertainty Management Systems. Kluwer Academic Publishers, Dordrecht, 1998.

    Google Scholar 

  24. Duentsch, I., Gediga, G.: Statistical evaluation of rough set dependency analysis. International Journal of Human-Computer Studies 46, 1997, pp. 589–604.

    Google Scholar 

  25. Duentsch, I., Gediga, G.: Rough set data analysis. In: Encyclopedia of Computer Science and Technology, Marcel Dekker (to appear).

    Google Scholar 

  26. Egenhofer, M.J., Golledge, R.G. (eds.): Spatial and Temporal Reasoning in Geographic Information Systems. Oxford University Press, Oxford, 1997.

    Google Scholar 

  27. Eisinger, M., Ohlbach, H. J.: Deduction systems based on resolution. In: Gabbay, D.M., Hogger, C.J., and Robinson, J.A. (eds.): Handbook of Logic in Artificial Intelligence and Logic Programming Vol.1. Oxford University Press, New York, 1993, pp. 183–271.

    Google Scholar 

  28. Fagin, R., Halpern, J.Y., Moses, Y., and Vardi, M.Y.: Reasoning about Knowledge. MIT Press, Cambridge MA, 1995.

    MATH  Google Scholar 

  29. Fayyad, U., Piatetsky-Shapiro, G. (eds.): Advances in Knowledge Discovery and Data Mining. MIT and AAAI Press, Cambridge MA, 1996.

    Google Scholar 

  30. Fitting, M.: Basic modal logic. In: Gabbay, D.M., Hogger, C.J., and Robinson, J.A. (eds.): Handbook of Logic in Artificial Intelligence and Logic Programming vol.3. Oxford University Press, New York, 1993, pp. 368–438.

    Google Scholar 

  31. Friedman, J., Kohavi, R., Yun, Y.: Lazy decision trees. In: Proc. of AAAI-96, 1996, pp. 717–724.

    Google Scholar 

  32. Gabbay, D., Guenthner, F. (eds.): Handbook of Philosophical Logic Vol.2. Kluwer Academic Publishers, Dordrecht, 1994.

    Google Scholar 

  33. Gabbay, D.M., Hogger, C.J., and Robinson, J.A. (eds.): Handbook of Logic in Artificial Intelligence and Logic Programming Vol.1. Oxford University Press, New York, 1993.

    MATH  Google Scholar 

  34. Gabbay, D.M., Hogger, C.J., and Robinson, J.A. (eds.): Handbook of Logic in Artificial Intelligence and Logic Programming vol.3. Oxford University Press, New York, 1993.

    Google Scholar 

  35. Gabbay, D.M., Hogger, C.J., and Robinson, J.A. (eds.): Handbook of Logic in Artificial Intelligence and Logic Programming Vol.4. Oxford University Press, New York, 1995.

    MATH  Google Scholar 

  36. Garey, M.R., Johnson, D.S.: Computers and Intractability. A Guide to the Theory of NP-completeness. W.H. Freeman and Company, New York, 1979.

    MATH  Google Scholar 

  37. Gärdenfors, P., Rott, H.: Belief revision. In: Gabbay, D.M., Hogger, C.J., and Robinson, J.A. (eds.): Handbook of Logic in Artificial Intelligence and Logic Programming Vol.4. Oxford University Press, New York, 1995, pp. 35–132.

    Google Scholar 

  38. Geffner, H.: Default Reasoning: Causal and Conditional theories. MIT Press, Cambridge, MA, 1992.

    Google Scholar 

  39. Gentzen, G.: Untersuchungen über das logische Schliessen, Math. Zeitschrift 39, 1934, pp. 176–210, 405–431.

    MathSciNet  Google Scholar 

  40. Gibson, J.I.: The Ecological Approach to Visual Perception, Hillsdale: Lawrence Erlbaum, 2nd edition, 1986.

    Google Scholar 

  41. Glymour, C., Madigan, D., Pregibon, D., Smyth, P.: Statistical themes and lessons for data mining. Data Mining and Knowledge Discovery 1, 1996, pp. 25–42.

    Google Scholar 

  42. Gonzalez, A.J., Dankel, D.D.: The Engineering of Knowledge Based Systems: Theory and Practice. Prentice Hall, Englewood Cliffs, NJ, 1993.

    MATH  Google Scholar 

  43. Gödel, K.: die Vollständigkeit der Axiome des Logischen Funktionenkalküls, Monatshefte für Mathematik und Physik 37, 1930, pp. 349–360.

    MATH  Google Scholar 

  44. Gödel, K.: Über formal unetscheidbare Sätze der Principia Mathematica und Verwandtersysteme I, Monatshefte für Mathematik und Physik 38, 1931, pp. 173–198.

    MATH  Google Scholar 

  45. Greco, S., Matarazzo, B., Slowinski, R.: Rough Approximation of a Preference Relation in a Pairwise Comparison Table. In: Polkowski, L., Skowron, A. (Eds.): Rough Sets in Knowledge Discovery 2: Applications, Case Studies and Software Systems. Physica-Verlag, Heidelberg, 1998, pp. 13–36.

    Google Scholar 

  46. Grzymala-Busse, J.W.: Applications of the rule induction system LERS. In: Polkowski, L., Skowron, A. (Eds.): Rough Sets in Knowledge Discovery 1: Methodology and Applications. Physica-Verlag, Heidelberg, 1998, pp. 366–375.

    Google Scholar 

  47. Halpern, J.Y.: Reasoning about knowledge: a survey. In: Gabbay, D.M., Hogger, C.J., and Robinson, J.A. (eds.): Handbook of Logic in Artificial Intelligence and Logic Programming Vol.4. Oxford University Press, New York, 1995, pp. 1–34.

    Google Scholar 

  48. Harel D.: Dynamic logic. In: Gabbay, D., Guenthner, F. (eds.): Handbook of Philosophical Logic Vol.2. Kluwer Academic Publishers, Dordrecht, 1994, pp. 497–604.

    Google Scholar 

  49. Hintikka, J.: Knowledge and Belief. Cornell University Press, Ithaca, N.Y., 1962.

    Google Scholar 

  50. Hodges, W.: Logical features of Horn clauses. In: Gabbay, D.M., Hogger, C.J., and Robinson, J.A. (eds.): Handbook of Logic in Artificial Intelligence and Logic Programming Vol.1. Oxford University Press, New York, 1993, 449–503.

    Google Scholar 

  51. Huber, P.J.: Robust statistics. Wiley, New York, 1981.

    MATH  Google Scholar 

  52. Hughes, G.E., Creswell, M.J.: An Introduction to Modal Logic. Methuen, London, 1968.

    MATH  Google Scholar 

  53. Komorowski, J., Żytkow, J. (Eds.): The First European Symposium on Principles of Data Mining and Knowledge Discovery (PKDD’97). June 25–27, Trondheim, Norway. Lecture Notes in Artificial Intelligence 1263, Springer-Verlag, Berlin, 1997, pp. 1–396.

    Google Scholar 

  54. Komorowski, J., Pawlak, Z., Polkowski, L., Skowron, A.: Rough sets: A tutorial. In: S.K. Pal and A. Skowron (eds.), Rough-Fuzzy Hybridization: A New Trend in Decision-Making. Springer-Verlag, Singapore, 1997, pp. 3–98.

    Google Scholar 

  55. Konar, A.: Artificial Intelligence and Soft Computing, Behavioral and Cognitive Modeling of the Human Brain, CRC Press, Boca Raton, 1999, pp. 49.

    Google Scholar 

  56. Konolige, K.: Autoepistemic logic. In: Gabbay, D.M., Hogger, C.J., and Robinson, J.A. (eds.): Handbook of Logic in Artificial Intelligence and Logic Programming vol.3. Oxford University Press, New York, 1994, 217–295.

    Google Scholar 

  57. Kowalczyk, W.: Rough data modelling, A new technique for analyzing data. In: Polkowski, L., Skowron, A. (Eds.): Rough Sets in Knowledge Discovery 1: Methodology and Applications. Physica-Verlag, Heidelberg, 1998, pp. 400–421.

    Google Scholar 

  58. Krawiec, K., Slowiński, R., Vanderpooten, D.: Learning decision rules from similarity based rough approximations. In: Polkowski, L., Skowron, A. (Eds.): Rough Sets in Knowledge Discovery 2: Applications, Case Studies and Software Systems. Physica-Verlag, Heidelberg, 1998, pp. 37–54.

    Google Scholar 

  59. Kruse, R., Gebhardt, J., and Klawonn, F.: Foundations of Fuzzy Systems. J. Wiley, New York, 1994.

    Google Scholar 

  60. Kryszkiewicz, M.: Generation of rules from incomplete information systems. In: Komorowski, J., Żytkow, J. (Eds.): The First European Symposium on Principles of Data Mining and Knowledge Discovery (PKDD’97). June 25–27, Trondheim, Norway. Lecture Notes in Artificial Intelligence 1263, Springer-Verlag, Berlin, 1997, pp. 156–166.

    Google Scholar 

  61. Leśniewski, S.: On the foundations of mathematics. In: Surma, S., Srzednicki, J.T., Barnett, D.I., Rickey, F.V. (eds), Stanislaw Leśniewski. Collected Works. Kluwer Academic Publishers, Dordrecht (1992), pp. 174–382.

    Google Scholar 

  62. Lifschitz, V.: Circumscription. In: Gabbay, D.M., Hogger, C.J., and Robinson, J.A. (eds.): Handbook of Logic in Artificial Intelligence and Logic Programming vol.3. Oxford University Press, New York, 1993, 297–352.

    Google Scholar 

  63. Lin, T.Y., Cercone, N. (Eds.): Rough Sets and Data Mining. Analysis of Imprecise Data. Kluwer Academic Publishers, Boston, 1997.

    MATH  Google Scholar 

  64. Lin, T.Y.: Granular computing on binary relations I, II. In: Polkowski, L., Skowron, A. (Eds.): Rough Sets in Knowledge Discovery 1: Methodology and Applications. Physica-Verlag, Heidelberg, 1998, pp. 107–140.

    Google Scholar 

  65. Liu, C., Zhong, N.: Rough problem settings for inductive logic programming, In: N. Zhong, A. Skowron, S. Ohsuga (eds.): New Directions in Rough Sets, Data Mining, and Granular — Soft Computing, Proceedings of the 7th International Workshop RSFDGSC’99, Yamaguchi, Japan, November 1999, Lecture Notes in Artificial Intelligence 1711, Springer-Verlag, Berlin, 1999, pp. 168–177.

    Google Scholar 

  66. Lloyd, J. W.: Foundations of Logic Programming. Springer- Verlag, Berlin, 1984.

    MATH  Google Scholar 

  67. Lukasiewicz, J.: Die logischen Grundlagen der Wahrscheinchkeitsrechnung. Krakow, 1913.

    Google Scholar 

  68. Lukasiewicz, J.: Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls. Comptes rendus de la Société des Sciences et des Lettres de Varsovie 23, 1930, pp. 57–77.

    Google Scholar 

  69. Makinson, D.: General patterns in non-monotonic reasoning. In: Gabbay, D.M., Hogger, C.J., and Robinson, J.A. (eds.): Handbook of Logic in Artificial Intelligence and Logic Programming vol.3. Oxford University Press, New York, 1993, 35–110.

    Google Scholar 

  70. Marek, V. W., Truszczyński, M.: Contributions to the theory of rough sets, Fundamenta Informaticae 39(4), 1999, pp. 389–409.

    MathSciNet  MATH  Google Scholar 

  71. Mendelson, E.: Introduction to Mathematical Logic. Van Nostrand, New York, 1960.

    Google Scholar 

  72. Michalski, R.: Inferential theory of learning as a conceptual basis for multistrategy. Machine Learning 11, 1993, pp. 111–151.

    MathSciNet  Google Scholar 

  73. Michalski R., Tecuci G.: Machine Learning. A Multistrategy Approach Vol.4. Morgan Kaufmann, San Francisco, 1994.

    Google Scholar 

  74. Michie, D., Spiegelhalter, D.J., Taylor, C.C. (Eds.): Machine Learning, Neural and Statistical Classification. Ellis Horwood, New York, 1994.

    MATH  Google Scholar 

  75. Mitchell, T.M.: Machine Learning. Mc Graw-Hill, Portland, 1997.

    MATH  Google Scholar 

  76. Mollestad, T., Komorowski, J.: A Rough Set Framework for Propositional Default Rules Data Mining. In: S.K. Pal and A. Skowron (Eds.): Rough-Fuzzy Hybridization: A New Trend in Decision Making. Springer-Verlag, Singapore, 1999.

    Google Scholar 

  77. Muggleton, S.: Foundations of Inductive Logic Programming. Prentice Hall, Englewood Cliffs, 1995.

    Google Scholar 

  78. Nadler M., Smith E.P.: Pattern Recognition Engineering. Wiley, New York, 1993.

    MATH  Google Scholar 

  79. Nguyen, H.S.: Discretization of Real Value Attributes, Boolean Reasoning Approach. Ph.D. Dissertation, Warsaw University, 1997, pp. 1–90.

    Google Scholar 

  80. Nguyen, H.S.: Efficient SQL-learning method for data mining in large data bases. In: Proceedings of the Sixteenth International Joint Conference on Artificial Intelligence (IJCAI’99), 1999, pp. 806–811.

    Google Scholar 

  81. Nguyen, S.H.: Data Regularity Analysis and Applications in Data Mining. Ph.D. Dissertation, Warsaw University, 1999.

    Google Scholar 

  82. Nguyen, H.S., Nguyen, S.H.: Pattern extraction from data. Fundamenta Informaticae 34, 1998, pp. 129–144.

    MathSciNet  MATH  Google Scholar 

  83. Nguyen, H.S., Nguyen, S.H.: Rough sets and association rule generation. Fundamenta Informaticae 40/4 (in print).

    Google Scholar 

  84. Nguyen, S.H., Skowron, A., Synak, P.: Discovery of data patterns with applications to decomposition and classification problems. In: Polkowski, L., Skowron, A. (Eds.): Rough Sets in Knowledge Discovery 2: Applications, Case Studies and Software Systems. Physica-Verlag, Heidelberg, 1998, pp. 55–97.

    Google Scholar 

  85. Nute, D.: Conditional logic. In: Gabbay, D., Guenthner, F. (eds.): Handbook of Philosophical Logic Vol.2. Kluwer Academic Publishers, Dordrecht, 1994, 387–440.

    Google Scholar 

  86. Orlowska, E. (ed.): Incomplete Information: Rough Set Analysis. Physica-Verlag, Heidelberg, 1998.

    Google Scholar 

  87. Pal, S.K., Skowron, A.: Rough-fuzzy Hybridization: A New Trend in Decision Making. Springer-Verlag, Singapore, 1999.

    MATH  Google Scholar 

  88. Paun, G., Polkowski, L., Skowron, A.: Parallel communicating grammar systems with negotiations. Fundamenta Informaticae 28/3–4, 1996, pp. 315–330.

    MathSciNet  MATH  Google Scholar 

  89. Pawlak, Z.: Information systems-theoretical foundations. Information Systems 6, 1981, pp. 205–218.

    MATH  Google Scholar 

  90. Pawlak, Z.: Rough sets. International Journal of Computer and Information Sciences 11, 1982, pp. 341–356.

    MathSciNet  MATH  Google Scholar 

  91. Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning about Data. Kluwer Academic Publishers, Dordrecht, 1992.

    Google Scholar 

  92. Pawlak, Z.: Decision Rules, Bayes’ Rule and Rough Sets, In: N. Zhong, A. Skowron, S. Ohsuga (eds.): New Directions in Rough Sets, Data Mining, and Granular — Soft Computing, Proceedings of the 7th International Workshop RSFDGSC’99, Yamaguchi, Japan, November 1999, Lecture Notes in Artificial Intelligence 1711, Springer-Verlag, Berlin, 1999. 1–9.

    Google Scholar 

  93. Pawlak, Z., Ras, Z. (Eds.): Proceedings: Ninth International Symposium on Methodologies for Intelligent Systems (ISMIS’96), Springer-Verlag, Berlin, 1996.

    Google Scholar 

  94. Pawlak, Z., Skowron, A.: Rough set rudiments. Bulletin of the International Rough Set Society 3/4, 1999, pp. 181–185.

    Google Scholar 

  95. Pearl, J.: Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Francisco, 1988.

    Google Scholar 

  96. Piasta, Z., Lenarcik, A.: Rule induction with probabilistic rough classifiers. Machine Learning (to appear).

    Google Scholar 

  97. Polkowski, L.: On synthesis of constructs for spatial reasoning via rough mereology. Fundamenta Informaticae (to appear).

    Google Scholar 

  98. Polkowski, L., Skowron, A.: Rough mereology: A new paradigm for approximate reasoning. International Journal of Approximate Reasoning 15/4,1994, pp. 333–365.

    MathSciNet  Google Scholar 

  99. Polkowski, L., Skowron, A.: Rough sets: A perspective. In: Polkowski, L., Skowron, A. (Eds.): Rough Sets in Knowledge Discovery 1: Methodology and Applications. Physica-Verlag, Heidelberg, 1998, pp. 31–58.

    Google Scholar 

  100. Polkowski, L., Skowron, A. (Eds.): Rough Sets in Knowledge Discovery 1: Methodology and Applications. Physica-Verlag, Heidelberg, 1998.

    MATH  Google Scholar 

  101. Polkowski, L., Skowron, A. (Eds.): Rough Sets in Knowledge Discovery 2: Applications, Case Studies and Software Systems. Physica-Verlag, Heidelberg, 1998.

    MATH  Google Scholar 

  102. Polkowski, L., Skowron, A. (Eds.): Proc. First International Conference on Rough Sets and Soft Computing, RSCTC’98, Warszawa, Poland, LNAI 1424, Springer-Verlag, Berlin, 1998.

    Google Scholar 

  103. Polkowski, L., Skowron, A.: Towards adaptive calculus of granules. In: Zadeh, L.A., Kacprzyk, J. (eds.): Computing with Words in Information/ Intelligent Systems Vol. 1–2. Physica-Verlag, Heidelberg, 1999, pp. 201–227.

    Google Scholar 

  104. Poole, D.: Default logic. In: Gabbay, D.M., Hogger, C.J., and Robinson, J.A. (eds.): Handbook of Logic in Artificial Intelligence and Logic Programming vol.3. Oxford University Press, New York, 1993, 189–216.

    Google Scholar 

  105. Prawitz, D.: Natural deduction, a proof theoretic study. Stockholm Studies in Philosophy Vol.3, Almquist & Wiksell, Stockholm, 1965.

    Google Scholar 

  106. Quinlan, J.R.: C4.5. Programs for machine learning. Morgan Kaufmann, San Mateo, CA, 1993.

    Google Scholar 

  107. Ras, ZW.: Cooperative knowledge-based systems. Journal of the Intelligent Automation Soft Computing 2/2 (special issue edited by T.Y. Lin), 1996, pp. 193–202.

    Google Scholar 

  108. Rescher, N.: Many-valued Logics. Mc Graw Hill, New York, 1969.

    Google Scholar 

  109. Rissanen, J.J.: Modeling by Shortest Data Description. Automatica 14, 1978, pp. 465–471.

    MATH  Google Scholar 

  110. Roddick J.F., Spiliopoulou M.: A Bibliography of Temporal, Spatial, and Temporal Data Mining Research. Newsletter of the Special Interest Group (SIG) on Knowledge Discovery & Data Mining 1/1, 1999, pp. 34–38.

    Google Scholar 

  111. Rosser, J.B., Turquette, A.R.: Many-valued Logics. North-Holland, Amsterdam, 1952.

    MATH  Google Scholar 

  112. Russel, S.J., Norvig, P.: Artificial Intelligence. A Modern Approach. Prentice Hall, Englewood Cliffs, 1995.

    Google Scholar 

  113. Sandewall, E., Shoham, Y.: Non-monotonic temporal reasoning. In: Gabbay, D.M., Hogger, C.J., and Robinson, J.A. (eds.): Handbook of Logic in Artificial Intelligence and Logic Programming Vol.4. Oxford University Press, New York, 1995, 439–498.

    Google Scholar 

  114. Selman, B., Kautz, H., McAllester, D.: Ten challenges in propositional reasoning and search. Proc. IJCAI’97, Japan.

    Google Scholar 

  115. Skowron, A.: Synthesis of adaptive decision systems from experimental data. In: A. Aamodt J. Komorowski (eds): Proc. of the Fifth Scandinavian Conference on Artificial Intelligence (SCAI’95), May 1995, Trondheim, Norway. IOS Press, Amsterdam, 1995, pp. 220–238.

    Google Scholar 

  116. Skowron, A., Nguyen, H.S.: Boolean reasoning scheme with some applications in data mining. Proceedings of the 3-rd European Conference on Principles and Practice of Knowledge Discovery in Databases, September 1999, Prague, Czech Republic. Lecture Notes in Computer Science 1704, 1999, pp. 107–115.

    Google Scholar 

  117. Skowron, A., Rauszer, C.: The Discernibility Matrices and Functions in Information Systems. In: Slowiński, R. (Ed.): Intelligent Decision Support — Handbook of Applications and Advances of the Rough Sets Theory. Dordrecht, Kluwer Academic Publishers, 1992, pp. 331–362.

    Google Scholar 

  118. Skowron, A., Stepaniuk, J.: Tolerance Approximation Spaces. Fundamenta Informaticae 27, 1996, pp. 245–253.

    MathSciNet  MATH  Google Scholar 

  119. Skowron, A., Stepaniuk, J., Tsumoto, S.: Information Granules for Spatial Reasoning. Bulletin of the International Rough Set Society 3/4, 1999, pp. 147–154.

    Google Scholar 

  120. Ślęzak, D.: Approximate reducts in decision tables. In: Proceedings of the Sixth International Conference, Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU’96) vol. 3, July 1–5, Granada, Spain, 1996, pp. 1159–1164.

    Google Scholar 

  121. D. Ślçzak: Decomposition and synthesis of approximate functional dependencies, PhD Thesis, Warsaw University (forthcoming).

    Google Scholar 

  122. Slowiński, R. (Ed.): Intelligent Decision Support — Handbook of Applications and Advances of the Rough Sets Theory. Dordrecht, Kluwer Academic Publishers, 1992.

    MATH  Google Scholar 

  123. Slowiriski, R., Vanderpooten, D.: Similarity relation as a basis for rough approximations. In: P. Wang (Ed.): Advances in Machine Intelligence & Soft Computing. Bookwrights, Raleigh NC, 1997, pp. 17–33.

    Google Scholar 

  124. Slowiński, R., Vanderpooten, D.: A generalized definition of rough approximations based on similarity. IEEE Trans. on Data and Knowledge Engineering (to appear).

    Google Scholar 

  125. http:agora.leeds.ac.ukspacenet.html

    Google Scholar 

  126. Stalnaker, R.: A theory of conditionals. In: N. Rescher (ed.): Studies in Logical Theory. Blackwell, Oxford, 1968.

    Google Scholar 

  127. Szczuka, M.: Symbolic and neural network methods for classifiers construction. Ph.D. Dissertation, Warsaw University, 1999.

    Google Scholar 

  128. Tarski, A.: On the concept of logical consequence. In: Logic, Semantics, Metamathematics. Oxford University Press, Oxford, 1956.

    Google Scholar 

  129. Tarski, A.: Der Wahrheitsbegriff in den Formalisierten Sprachen, Studia Philosophica 1, 1936, pp. 261–405.

    Google Scholar 

  130. Troelstra, A. S.: Aspects of constructive mathematics. In: Barwise, J. (ed.): Handbook of Mathematical Logic. North Holland, Amsterdam, 1977, pp. 973–1052.

    Google Scholar 

  131. Tsang, E.: Foundations of Constraint Satisfaction. Academic Press, London 1993.

    Google Scholar 

  132. Tsumoto, S.: Modelling diagnostic rules based on rough sets. In: Polkowski, L., Skowron, A. (Eds.): Proc. First International Conference on Rough Sets and Soft Computing, RSCTC’98, Warszawa, Poland, LNAI 1424, Springer-Verlag, Berlin, 1998, pp. 475–482.

    Google Scholar 

  133. Ullman, J.D., Widom, J.: A First Course in Database Systems. Prentice—Hall, Inc., Englewood Cliffs, 1997.

    Google Scholar 

  134. Urquhart, A.: Many-valued logic. In: Gabbay, D.M., Hogger, C.J., and Robinson, J.A. (eds.): Handbook of Logic in Artificial Intelligence and Logic Programming vol.3. Oxford University Press, New York, 1993, 71–116.

    Google Scholar 

  135. Wasilewska, A., Vigneron, L.: Rough algebras and automated deduction. In: Polkowski, L., Skowron, A. (Eds.): Rough Sets in Knowledge Discovery 1: Methodology and Applications. Physica-Verlag, Heidelberg, 1998, pp. 261–275.

    Google Scholar 

  136. Wille, R.: Formale Begriffsanalyse: Mathematische Grundlagen. Springer-Verlag, Berlin, 1996.

    MATH  Google Scholar 

  137. Von Wright, G.H.: An Essay in Modal Logic. North Holland, Amsterdam, 1951.

    MATH  Google Scholar 

  138. Zadeh, L.A.: Fuzzy sets. Information and Control 8, 1965, pp. 333–353.

    MathSciNet  Google Scholar 

  139. Zadeh, L.A.: A theory of approximate reasoning. In: Hayes, J.E., Michie, D., Mikulich, L.C. (eds.): Machine Intelligence Vol.9. J. Wiley, New York, 1979, pp. 149–194.

    Google Scholar 

  140. Zadeh, L.A.: Fuzzy logic = computing with words. IEEE Trans. on Fuzzy Systems 4, 1996, pp. 103–111.

    Google Scholar 

  141. Zadeh, L.A: Toward a theory of fuzzy information granulation and its certainty in human reasoning and fuzzy logic. Fuzzy Sets and Systems 90, 1997, pp. 111–127.

    MathSciNet  MATH  Google Scholar 

  142. Zadeh, L.A., Kacprzyk, J. (eds.): Computing with Words in Information/ Intelligent Systems Vol. 1–2. Physica-Verlag, Heidelberg, 1999.

    Google Scholar 

  143. Zaki, M.J., Parthasarathy, S., Ogihara, M., Li, W.: New parallel algorithms for fast discovery of association rules. In: Data Mining and Knowledge Discovery : An International Journal (special issue on Scalable High-Performance Computing for KDD) 1/4, 1997, pp. 343–373.

    Google Scholar 

  144. Ziarko, W.: Variable Precision Rough Set Model. J. of Computer and System Sciences 46, 1993, pp. 39–59.

    MathSciNet  MATH  Google Scholar 

  145. Ziarko, W. (ed.): Rough Sets, Fuzzy Sets and Knowledge Discovery (RSKD’93). Workshops in Computing, Springer-Verlag & British Computer Society, London, Berlin, 1994.

    Google Scholar 

  146. Ziarko, W.: Rough sets as a methodology for data mining. In: Polkowski, L., Skowron, A. (Eds.): Rough Sets in Knowledge Discovery 2: Applications, Case Studies and Software Systems. Physica-Verlag, Heidelberg, 1998, pp. 554–576.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Information Technology and Management, Newelska 6, 01-447, Warsaw, Poland

    Zdzisław Pawlak

  2. Polish-Japanese Institute of Information Technology, Koszykowa 86, 02-008, Warszawa, Poland

    Lech Polkowski

  3. Department of Mathematics and Information Sciences, Warsaw University of Technology, Pl. Politechniki 1, 00-665, Warszawa, Poland

    Lech Polkowski

  4. Institute of Mathematics Warsaw University, Banacha 2, 02-097, Warszawa, Poland

    Andrzej Skowron

Authors
  1. Zdzisław Pawlak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lech Polkowski
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andrzej Skowron
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics and Information Sciences, Warsaw University of Technology, Pl. Politechnik 1, 00650, Warsaw, Poland

    Lech Polkowski

  2. Department of Medical Informatics, Shinmane Medical University, 89-1 Enya-cho, 693-8501, Izumo City, Shimane, Japan

    Shusaku Tsumoto

  3. Department of Mathematics and Computer Science, San Jose State University, San Jose, CA, 95192, USA

    Tsau Y. Lin

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Physica-Verlag Heidelberg

About this chapter

Cite this chapter

Pawlak, Z., Polkowski, L., Skowron, A. (2000). Rough Sets and Rough Logic: A KDD Perspective. 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_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-7908-1840-6_13

  • Publisher Name: Physica, Heidelberg

  • Print ISBN: 978-3-662-00376-3

  • Online ISBN: 978-3-7908-1840-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation