An investigation about rough set theory: Some foundational and mathematical aspects

Fundamenta Informaticae

Volumn 108, Issue 3-4, 2011, Pages 197-221

  Cattaneo, Gianpiero ^a  

^a UNIVERSITY OF MILANO BICOCCA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION SPACES; COMPLETE INFORMATION; EQUIVALENCE RELATIONS; FORMAL ANALYSIS; INCOMPLETE INFORMATION SYSTEMS; LOWER AND UPPER APPROXIMATIONS; MATHEMATICAL ASPECTS; MATHEMATICAL STRUCTURE; SIMILARITY RELATIONS;

INFORMATION SYSTEMS;

ROUGH SET THEORY;

EID: 80052690507     PISSN: 01692968     EISSN: None     Source Type: Journal    
DOI: 10.3233/FI-2011-420     Document Type: Conference Paper

References (61)

1. P. Alexandrov, Discrete r̈aume, Mathematiceskij Sbornik 2 (1937), 501-518.
2. F. G. Arenas, Alexandroff spaces, Acta Math. Univ. Comenianae (1999), 17-25.
3. W. Bartol, J. Miŕo, K. Píoro, and F. Rossello, On the covering by tolerance classes, Information Sciences 166 (2004), 193-211.
4. R. Carnap, Der logische aufbau der welt, Weltkreis-Verlag, 1928.
5. G. Cattaneo, Abstract approximation spaces for rough theories, in Polkowski and Skowron [41], pp. 59-98.
6. G. Cattaneo, The lattice topological approach to roughness theory by approximation spaces, in Proc. International Conference on Rough Sets, Fuzzy sets and Soft Computing (R. N. Bhaumik ed. ), Department of Mathematics, Tripura University (India), 2011, pp. 9-74.
7. G. Cattaneo and D. Ciucci, A hierarchical lattice closure approach to abstract approximation spaces, Lecture Notes in Artificial Intelligence, vol. 5009, 2008, pp. 363-370. (Pubitemid 351776048)
8. G. Cattaneo and D. Ciucci, Lattice with interior and closure operators and abstract approximation spaces, Foundations of Rough Sets (J. F. Peters et al., ed. ), LNCS-Transactions on Rough Sets X, Springer, 2009, pp. 67-116.
9. G. Cattaneo and D. Ciucci, About some different topological approaches to covering rough sets, pre-print submitted for publication, 2010.
10. B. F. Chellas, Modal logic, an introduction, Cambridge University Press, Cambridge, MA, 1988, (first published 1980).
11. M. Erńe, Closure, ContemporaryMathematics 486 (2009), 163-238.
12. C. J. Everett, Closure operators and Galois theory in lattices, Transaction of the American Mathematical Society 55 (1944), 514-525.
13. J. W. Grzymala-Busse, On the unknown attribute values in learning from examples, Lecture Notes in Artificial Intelligence, vol. 542, 1991, pp. 368-377.
14. P. R. Halmos, The basic concepts of algebraic logic, American Mathematical Monthly 53 (1956), 363-387.
15. P. R. Halmos, Algebraic logic, Chelsea Pub. Co., New York, 1962.
16. K. Iseki, On closure operation in lattice theory, Nederland Akad. Wetensch. Proc. Ser. A 54; Indag. Math. 3 (1951), 318-320.
17. T. B. Iwinski, Algebraic approach for rough sets, Bulletin of the Polish Academy of Sciences-Mathematics 35 (1987), 673-683.
18. J. L. Kelley, General topology, Springer-Verlag, New York, 1955, (second edition 1957 by Van Nostrand).
19. M. Kryszkiewicz, Rough set approach to incomplete information systems, Proceeding of Second Annual Join Conference on Information Sciences (Wrightsville Beach, North Carolina, USA), 28 September-1 Octobe 1995 1995, pp. 194-197.
20. M. Kryszkiewicz, generation of rules from incomplete information systems, Lecture Notes in Artificial Intelligence, vol. 1263, 1997, pp. 156-166. (Pubitemid 127097501)
21. M. Kryszkiewicz, Rough set approach to incomplete information systems, Information Sciences 112 (1998), 39-49. (Pubitemid 128649627)
22. C. Kuratowski, Sur l'oṕeration A de l'analysis situs, FundamentaMathematicae 3 (1922), 182-199.
23. C. Kuratowski, Topologie. I, MonografieMatematyczne, Warszawa, 1933, (Other editions 1948 and 1958).
24. T. Y. Lin and A. M. Wildberger (eds. ), Rough sets and soft computing, third international workshop (RSSC94), The Society for Computer Simulation, San Jose, California, 1994.
25. S. Marcus, Tolerance rough sets, Cech topologies, learnig processes, Bulletin of the Polish Academy of Sciences-Technical Sciences 42 (1994), 171-187.
26. A. Monteiro, Caracterisation de l'operation de fermeture par un seul axiome, PortugaliaeMathematica 4 (1945), 158-160.
27. A. Monteiro and H. Ribeiro, L'operation de fermeture et ses invariants dans les systemes partiellement ordennes, PortugaliaeMathematica 3 (1942), 171-184.
28. O. Ore, Combinations of closure relations, Annals of Mathematics 44 (1942), 514-533.
29. O. Ore, Galois connexions, Transactions of the American Mathematical Society 55 (1944), 493-513.
30. Z. Pawlak, Information systems-theoretical foundations, Information Systems 6 (1981), 205-218.
31. Z. Pawlak, Rough sets, Int. J. of Computer and Information Sciences 11 (1982), 341-356.
32. Z. Pawlak, On rough sets, Bulletin of the EATCS (1984), 94-109.
33. Z. Pawlak, Rough sets and decision tables, Lecture Notes in Computer Sciences, vol. 208, 1985, pp. 177-186.
34. Z. Pawlak, Rough logic, Bulletin of the Polish Academy of Sciences-Technical Sciences 35 (1987), 253-258.
35. Z. Pawlak, Rough sets-theoretical aspects of reasoning about data, Kluwer Academic Publishers, Dordrecht, 1991.
36. Z. Pawlak, Rough sets: A new approach to vagueness, Fuzzy Logic for the Management of Uncertainty (L. A. Zadeh and J. Kacprzyc, eds. ), J. Wiley and Sons, New York, 1992, pp. 105-118.
37. Z. Pawlak, Rough set elements, in Polkowski and Skowron [41], pp. 10-30.
38. H. Poincaŕe, Le continu math́ematique, Revue de Ḿetaphysique et de Morale I (1893), 26-34, (Reprinted in as Chapter II)
39. H. Poincaŕe, La science et l'hypoth̀ese, Flammarion, Paris, 1903, (English translation as Science and Hypothesis, Dover, New York, 1952).
40. J. A. Pomykala, Aproximation operations in approximation space, Bulletin of the Polish Academy of Sciences-Mathematics 35 (1987), 653-662.
41. L. Polkowski and A. Skowron (eds. ), Rough sets in knowledge discovery 1, Physica-Verlag, Heidelberg, New York, 1998.
42. L. Polkowski, A. Skowron, and J. Zytkow, Tolerance based rough sets, in Lin andWildberger [24], pp. 55-58.
43. G. F. Simmons, Topology and modern analysis, McGraw-Hill Book Company, Inc., NY, 1963.
44. A. Skowron, On topology in information systems, Bulletin of the polish Academy of Sciences-Mathematics 36 (1988), 477-479.
45. M. B. Smyth, Topology, Handbook of Logic in Computer Science (S. Abramsky, D. Gabbay, and T. Maibaum, eds. ), Oxford U. P., 1992, pp. 641-761.
46. R. Slowinski and J. Stefanowski, Rough classification in incomplete information systems, Mathematical and Computer Modelling 12 (1989), 1347-1357. (Pubitemid 20669867)
47. A. Skowron and J. Stepaniuk, Generalized approximation spaces, Rough Sets and Soft Computing, Third InternationalWorkshop (RSSC94), 1994, pp. 55-58.
48. R. Slowinski and J. Stefanowski, Handling various types of uncertainty in the rough set approach, Rough Sets, fuzzy sets and knowledge discovery. Workshop in Computer Serie (W. Ziarko, ed. ), Springer-Verlag, 1994, pp. 366-376.
49. R. Slowinski and J. Stefanowski, Rough-set reasoning about uncertain data, Fundamenta Informaticae 27 (1996), 229-243. (Pubitemid 126715033)
50. J. Stefanowki and A. Tsoukìas, On the extension of rough sets under incomplete information, Lecture Notes in Artificial Intelligence, vol. 1711, 1999, pp. 73-81.
51. M. Schroeder and M. Wright, Tolerance and weak tolerance relations, Journal of Combinatorial Mathematics and Combinatorial Computing 11 (1992), 123-160.
52. A. Tarski, Fundamentale Begriffe der Methodologie der deduktiven Wissennschaften. I, Monathshefte fur Mathematik und Physik 37 (1930), 361-404, (English version in [53]).
53. A. Tarski, Logic, semantics, metamathematics, Hackett, Indianapolis, 1983, (Second Edition-First Edition, 1956 by Oxford).
54. E. Cech, Topological spaces, Interscience Publisher, London/New york, 1966, revised edition.
55. M. Ward, The closure operator on lattice, Annals of Mathematics 43 (1942), 191-196.
56. U. Wybraniec-Skardowska, On a generalization of approximation space, Bulletin of the Polish Academy of Sciences-Mathematics 37 (1989), 51-61.
57. Y. Y. Yao, Relational interpretations of neighborhood operators and rough set approximation operators, Information Sciences 111 (1998), 239-259. (Pubitemid 128383897)
58. Y. Y. Yao, On generalizing rough set theory, Lecture Notes in Artificial Intelligence, vol. 2639, 2003.
59. W. Zakowski, Approximations in the space (U), Demonstration Mathematica XVI (1983), 761-769.
60. E. C. Zeeman, The topology of the brain and visual perception, Topology of 3-manifolds and related topics, Prentice-Hall, Englewood Cliffs, 1962, pp. 240-256.
61. W. Zhu, Toplogical approaches to covering rough sets, Information Sciences 177 (2007), 1499-1508. (Pubitemid 46037660)