A Framework for Reasoning with Rough Sets

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3700)


Rough sets framework has two appealing aspects. First, it is a mathematical approach to deal with vague concepts. Second, rough set techniques can be used in data analysis to find patterns hidden in the data. The number of applications of rough sets to practical problems in different fields demonstrates the increasing interest in this framework and its applicability. This thesis proposes a language that caters for implicit definitions of rough sets obtained by combining different regions of other rough sets. In this way, concept approximations can be derived by taking into account domain knowledge. A declarative semantics for the language is also discussed. It is then shown that programs in the proposed language can be compiled to extended logic programs under the paraconsistent stable model semantics. The equivalence between the declarative semantics of the language and the declarative semantics of the compiled programs is proved. This transformation provides the computational basis for implementing our ideas. A query language for retrieving information about the concepts represented through the defined rough sets is also discussed. Several motivating applications are described. Finally, an extension of the proposed language with numerical measures is presented. This extension is motivated by the fact that numerical measures are an important aspect in data mining applications.


Decision Rule Logic Program Integrity Constraint Decision Table Predicate Symbol 
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.


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  1. 1.
    Pawlak, Z.: Rough sets. International Journal of Information and Computer Science 11, 341–356 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Nguyen, H.S., Nguyen, T.T., Skowron, A., Synak, P.: Knowledge discovery by rough set methods. In: Callaos, N.C. (ed.) Proc. of the International Conference on Information Systems Analysis and Synthesis (ISAS 1996), pp. 526–533 (1996)Google Scholar
  3. 3.
    Komorowski, J., Øhrn, A.: Modelling prognostic power of cardiac tests using rough sets. Journal of Artificial Intelligence in Medicine 15, 167–191 (1999)CrossRefGoogle Scholar
  4. 4.
    Lazareck, L., Ramanna, S.: Classification of swallowing sound signals: A rough set approach. In: Tsumoto, S., Słowiński, R., Komorowski, J., Grzymała-Busse, J.W. (eds.) RSCTC 2004. LNCS (LNAI), vol. 3066, pp. 679–684. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  5. 5.
    Tay, F.E.H., Shen, L.: Economic and finantial prediction using rough sets model. European Journal of Operational Research 141, 641–659 (2002)zbMATHCrossRefGoogle Scholar
  6. 6.
    Midelfart, H., Komorowski, J.: A rough set framework for learning in a directed acyclic graph. In: Alpigini, J.J., Peters, J.F., Skowron, A., Zhong, N. (eds.) RSCTC 2002. LNCS (LNAI), vol. 2475, pp. 144–155. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  7. 7.
    Pagliani, P.: Rough sets theory and logic-algebraic structures. In: Orlowska, E. (ed.) Incomplete Information: Rough Sets Analysis, pp. 109–190. Physics (1997)Google Scholar
  8. 8.
    Yao, Y.Y., Lin, T.Y.: Generalizations of rough sets using modal logic. Journal of Intelligent Automation and Soft Computing 2, 103–120 (1996)Google Scholar
  9. 9.
    Midelfart, H., Komorowski, J.: A rough set approach to inductive logic programming. In: Ziarko, W.P., Yao, Y.Y. (eds.) RSCTC 2000. LNCS (LNAI), vol. 2005, p. 190. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  10. 10.
    Wygralak, M.: Rough sets and fuzzy sets - some remarks on interrelations. Journal of Fuzzy Sets and Systems 29, 241–243 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dubois, D., Prade, H.: Putting rough sets and fuzzy sets together. In: Slowinski, R. (ed.) Handbook of Applications and Advances of the Rough Sets Theory, pp. 204–232. Kluwer Academic Publishers, Dordrecht (1992)Google Scholar
  12. 12.
    Skowron, A., Stepaniuk, J.: Tolerance approximation spaces. Fundamenta Informaticae 27, 245–253 (1996)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Ziarko, W.: Variable precision rough set model. Journal of Computer and Systems Science 46, 39–59 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Øhrn, A., Komorowski, J.: ROSETTA: A rough set toolkit for analysis of data. In: Proc. of Third International Joint Conference on Information Sciences, Fifth International Workshop on Rough Sets and Soft Computing (RSSC 1997), Durham, NC, USA, vol. 3, pp. 403–407 (1997)Google Scholar
  15. 15.
    Ziarko, W., Fei, X.: VPRSM approach to WEB searching. In: Alpigini, J.J., Peters, J.F., Skowron, A., Zhong, N. (eds.) RSCTC 2002. LNCS (LNAI), vol. 2475, pp. 514–521. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  16. 16.
    Doherty, P., Łukaszewicz, W., Szałas, A.: CAKE: A Computer Aided Knowledge Engineering Technique. In: van Harmelen, F. (ed.) Proc. of the 15th European Conference on Artificial Intelligence (ECAI 2002), pp. 220–224. IOS Press, Amsterdam (2002)Google Scholar
  17. 17.
    Małuszyński, J., Vitória, A.: Defining rough sets by extended logic programs. In: On-Line Proc. of Paraconsistent Computational Logic Workshop, PCL 2002 (2002),,
  18. 18.
    Vitória, A., Małuszyński, J.: A logic programming framework for rough sets. In: Alpigini, J.J., Peters, J.F., Skowron, A., Zhong, N. (eds.) RSCTC 2002. LNCS (LNAI), vol. 2475, pp. 205–212. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  19. 19.
    Vitória, A., Damásio, C.V., Małuszyński, J.: Query answering for rough knowledge bases. In: Wang, G., Liu, Q., Yao, Y., Skowron, A. (eds.) RSFDGrC 2003. LNCS (LNAI), vol. 2639, pp. 197–204. Springer, Heidelberg (2003)Google Scholar
  20. 20.
    Damásio, C.V., Pereira, L.M.: A survey of paraconsistent semantics for logic programs. In: Gabbay, D.M., Smets, P. (eds.) Handbook of Defeasible Reasoning and Uncertainty Management Systems, vol. 2, pp. 241–320. Kluwer Academic Publishers, Dordrecht (1998)Google Scholar
  21. 21.
    Vitória, A., Damásio, C.V., Małuszyński, J.: From rough sets to rough knowledge bases. Fundamenta Informaticae 57, 215–246 (2003)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Vitória, A., Damásio, C.V., Małuszyński, J.: Toward rough knowledge bases with quantitative measures. In: Tsumoto, S., Słowiński, R., Komorowski, J., Grzymała-Busse, J.W. (eds.) RSCTC 2004. LNCS (LNAI), vol. 3066, pp. 153–158. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  23. 23.
    Andersson, R.: Rough Knowledge Base System (2004), Available at,
  24. 24.
    Pawlak, Z.: Rough sets. Theoretical Aspects of Reasoning about Data. Kluwer Academic Publishers, Dordrecht (1991)zbMATHGoogle Scholar
  25. 25.
    Komorowski, J., Pawlak, Z., Polkowski, L., Skowron, A.: Rough sets: A tutorial. In: Pal, S.K., Skowron, A. (eds.) Rough Fuzzy Hybridization. A New Trend in Decision-Making, pp. 3–98. Springer, Heidelberg (1999)Google Scholar
  26. 26.
    Ziarko, W.: Rough set approaches for discovery of rules and attribute dependencies. In: Kloesgen, W., Zytkow, J. (eds.) Handbook of Data Mining and Knowledge Discovery, pp. 328–338. Oxford University Press, Oxford (2002)Google Scholar
  27. 27.
    Slowinski, R., Vanderpooten, D.: Similarity relations as a basis for rough set approximations. In: Wang, P.P. (ed.) Advances in Machine Intelligence and Soft Computing, vol. 4, pp. 17–33 (1997)Google Scholar
  28. 28.
    Skowron, A., Polkowski, L.: Synthesis of decison systems from data tables. In: Lin, T.Y., Cercone, N. (eds.) Rough Sets and Data Mining Analysis of Imprecise Data, pp. 259–300. Kluwer Academic Publishers, Dordrecht (1997)Google Scholar
  29. 29.
    Bazan, J.G.: Discovery of decision rules by matching new objects against data tables. In: Polkowski, L., Skowron, A. (eds.) RSCTC 1998. LNCS (LNAI), vol. 1424, pp. 521–528. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  30. 30.
    Stefanowski, J.: On rough set approaches to induction of decision rules. In: Polkowski, L., Skowron, A. (eds.) Rough Sets in Knowledge Discovery, Methodology and Applications, pp. 501–529. Springer, Heidelberg (1998)Google Scholar
  31. 31.
    Bazan, J.G.: Dynamic reducts and statistical inference. In: Proc. of the Sixth International Conference, Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMIU 1996), vol. 3, pp. 1147–1152 (1996)Google Scholar
  32. 32.
    Lloyd, J.W.: Foundations of Logic Programming. Springer, Heidelberg (1987)zbMATHGoogle Scholar
  33. 33.
    Nilsson, U., Małuszynski, J.: Logic, Programming and Prolog, 2nd edn. John Wiley & Sons, Chichester (1995), Google Scholar
  34. 34.
    Baral, C.: Knowledge Representation, Reasoning and Declarative Problem Solving. Cambridge University Press, Cambridge (2003)zbMATHCrossRefGoogle Scholar
  35. 35.
    Brachman, R.J., Levesque, H.J.: Knowledge Representation and Reasoning. Elsevier, Amsterdam (2004)Google Scholar
  36. 36.
    Pearce, D.: Answer sets and constructive logic, II: Extended logic programs and related non-monotonic formalisms. In: Pereira, L.M., Nerode, A. (eds.) Logic Programming and Nonmonotonic Reasoning - Proc. of the 2nd International Workshop, pp. 457–475. MIT Press, Cambridge (1993)Google Scholar
  37. 37.
    Sakama, C., Inoue, K.: Paraconsistent Stable Semantics for Extended Disjunctive Programs. Journal of Logic and Computation 5, 265–285 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Gelfond, M., Lifschitz, V.: Logic programs with classical negation. In: Warren, S. (ed.) Proc. of the Seventh International Conference on Logic Programming, pp. 579–597. MIT Press, Cambridge (1990)Google Scholar
  39. 39.
    Gelfond, M., Lifschitz, V.: The stable model semantics for logic programming. In: Kowalski, R.A., Bowen, K. (eds.) Proc. of the Fifth International Logic Programming Conference and Symposium, Seattle, USA, pp. 1070–1080. MIT Press, Cambridge (1988)Google Scholar
  40. 40.
    Inoue, K., Sakama, C.: Negation as failure in the head. Journal of Logic Programming 35, 39–78 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Alferes, J.J., Leite, J.A., Pereira, L.M., Przymusinska, H., Przymusiski, T.C.: Dynamic updates of non-monotonic knowledge bases. Journal of Logic Programming. 45, 43–70 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Niemelä, I., Simons, P.: Efficient implementation of the well-founded and stable model semantics. In: Maher, M. (ed.) Proc. of the Joint International Conference and Symposium on Logic Programming, Bonn, Germany, pp. 289–303. MIT Press, Cambridge (1996)Google Scholar
  43. 43.
    Simons, P.: Smodels system, Available at,
  44. 44.
    Eiter, T., Leone, N., Mateis, C., Pfeifer, G., Scarcello, F.: The KR system dlv: Progress report, comparisons and benchmarks. In: Cohn, A.G., Schubert, L., Shapiro, S.C. (eds.) KR 1998: Principles of Knowledge Representation and Reasoning, pp. 406–417. Morgan Kaufmann, San Francisco (1998)Google Scholar
  45. 45.
    The dlv project, Available at,
  46. 46.
    Deransart, P., Ed-Bali, A., Cervoni, L.: Prolog: The Standard Reference Manual. Springer, Heidelberg (1996)zbMATHGoogle Scholar
  47. 47.
    Belnap, N.D.: A useful four-valued logic. In: Epstein, G., Dunn, J.M. (eds.) Modern Uses of Multiple-Valued Logic, pp. 7–37. Reidel Publishing Company, Dordrecht (1977)Google Scholar
  48. 48.
    Belnap, N.D.: How computer should think. In: Rydle, G. (ed.) Contemporary Aspets of Philosophy, pp. 30–56. Oriel Press (1977)Google Scholar
  49. 49.
    Niemelä, I., Simons, P.: Smodels - an implementation of stable model and the well-founded semantics for normal logic programs. In: Fuhrbach, U., Dix, J., Nerode, A. (eds.) LPNMR 1997. LNCS(LNAI), vol. 1265, pp. 420–429. Springer, Heidelberg (1997)Google Scholar
  50. 50.
    Dantsin, E., Eiter, T., Gottlob, G., Voronkov, A.: Complexity and expressive power of logic programming. ACM Computing Surveys 33, 374–425 (2001)CrossRefGoogle Scholar
  51. 51.
    Ziarko, W.: Acquisition of Hierarchy-Structured Probabilistic Decision Tables and Rules from Data. In: Proc. of the World Congress on Computational Intelligence, Honolulu (2002)Google Scholar
  52. 52.
    Reiter, R.: A logic for deafult reasoning. Journal of Artificial Intelligence 13, 81–132 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    Andersson, R.: Implementation of a rough knowledge base system supporting quantitative measures. Master thesis, Linköping University, IDA (2004)Google Scholar
  54. 54.
    Zaniolo, C.: Key constraints and monotonic aggregates in deductive databases. In: Kakas, A.C., Sadri, F. (eds.) Computational Logic: Logic Programming and Beyond. LNCS (LNAI), vol. 2408, pp. 109–134. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  55. 55.
    Faber, W., Leone, N., Pfeifer, G.: Recursive aggregates in disjunctive logic programs: Semantics and complexity. In: Alferes, J.J., Leite, J. (eds.) JELIA 2004. LNCS (LNAI), vol. 3229, pp. 200–212. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  56. 56.
    Java. Sun Mycrosystems, Available at,
  57. 57.
    XSB system, Available at,
  58. 58.
    Chen, W., Swift, T., Warren, D.S.: Efficient top-down computation of queries under the well-founded semantics. Journal of Logic Programming 24, 161–199 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  59. 59.
    Chen, W., Warren, D.S.: Tabled evaluation with delaying for general logic programs. Journal of the ACM 43, 20–74 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  60. 60.
    Sagonas, K., Swift, T.: An abstract machine for tabled execution of fixed-order stratified logic programs. Journal of the ACM TOPLAS 20, 586–635 (1998)CrossRefGoogle Scholar
  61. 61.
    Swift, T.: Tabling for non-monotonic reasoning. Annals of Mathematics and Artifcial Intelligence 25, 201–240 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  62. 62.
    Doherty, P., Kachniarz, J., Szałas, A.: Using contextually closed queries for local closed-world reasoning in rough knowledge databases. In: Pal, S.K., Polkowski, L., Skowron, A. (eds.) Rough-Neural Computing, pp. 219–250. Springer, Heidelberg (2004)Google Scholar
  63. 63.
    Polkowski, L., Skowron, A.: Rough mereology. In: Proc. of the Eighth International Symposium on Methodologies for Intelligent Systems, pp. 85–94. Springer, Heidelberg (1994)Google Scholar
  64. 64.
    Polkowski, L., Skowron, A.: Rough mereology: a new paradigm for approximate reasoning. International Journal of Approximate Reasoning 15, 333–365 (1997)CrossRefMathSciNetGoogle Scholar
  65. 65.
    Polkowski, L., Skowron, A.: Rough mereological calculi of granules: A rough set approach to computation. Journal of Computational Intelligence 17, 472–492 (2001)CrossRefMathSciNetGoogle Scholar
  66. 66.
    Midelfart, H., Lægreid, A., Komorowski, J.: Classification of gene expression data in an ontology. In: Crespo, J.L., Maojo, V., Martin, F. (eds.) ISMDA 2001. LNCS, vol. 2199, pp. 186–194. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  67. 67.
    Midelfart, H.: Knowledge Discovery from cDNA Microarrays and a priori Knowledge. PhD thesis, Department of Computer and Information Science, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Dept. of Science and TechnologyLinköping UniversityNorrköpingSweden

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