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Decision Trees with at Most 19 Vertices for Knowledge Representation Read chapter Read first chapter

2020 | OriginalPaper | Chapter

A Study of Algebras and Logics of Rough Sets Based on Classical and Generalized Approximation Spaces

Author : Arun Kumar

Published in: Transactions on Rough Sets XXII

Publisher: Springer Berlin Heidelberg

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Abstract

The seminal work of Z. Pawlak [60] on rough set theory has attracted the attention of researchers from various disciplines. Algebraists introduced some new algebraic structures and represented some old existing algebraic structures in terms of algebras formed by rough sets. In Logic, the rough set theory serves the models of several logics. This paper is an amalgamation of algebras and logics of rough set theory.

We prove a structural theorem for Kleene algebras, showing that an element of a Kleene algebra can be looked upon as a rough set in some appropriate approximation space. The proposed propositional logic \(\mathcal {L}_{K}\) of Kleene algebras is sound and complete with respect to a 3-valued and a rough set semantics.

This article also investigates some negation operators in classical rough set theory, using Dunn’s approach. We investigate the semantics of the Stone negation in perp frames, that of dual Stone negation in exhaustive frames, and that of Stone and dual Stone negations with the regularity property in \(K_{-}\) frames. The study leads to new semantics for the logics corresponding to the classes of Stone algebras, dual Stone algebras, and regular double Stone algebras. As the perp semantics provides a Kripke type semantics for logics with negations, exploiting this feature, we obtain duality results for several classes of algebras and corresponding frames.

In another part of this article, we propose a granule-based generalization of rough set theory. We obtain representations of distributive lattices (with operators) and Heyting algebras (with operators). Moreover, various negations appear from this generalized rough set theory and achieved new positions in Dunn’s Kite of negations.

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previous chapter Sequences of Refinements of Rough Sets: Logical and Algebraic Aspects
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Literature
  1. Avron, A., Konikowska, B.: Rough sets and 3-valued logics. Studia Logica 90, 69–92 (2008)MathSciNetMATHView Article
  2. Balbes, R., Dwinger, P.: Distributive Lattices. University of Missouri Press, Columbia (1974)MATH
  3. Banerjee, M.: Rough sets and 3-valued Łukasiewicz logic. Fundamenta Informaticae 31, 213–220 (1997)MathSciNetMATHView Article
  4. Banerjee, M., Chakraborty, M.K.: Rough sets through algebraic logic. Fundamenta Informaticae 28(3–4), 211–221 (1996)MathSciNetMATHView Article
  5. Banerjee, M., Chakraborty, M.K.: Algebras from rough sets. In: Pal, S.K., Polkowski, L., Skowron, A. (eds.) Rough-neuro Computing: Techniques for Computing with Words, pp. 157–184. Springer, Berlin (2004). https://​doi.​org/​10.​1007/​978-3-642-18859-6_​7View Article
  6. Banerjee, M., Khan, M.: Propositional logics from rough set theory. In: Peters, J.F., Skowron, A., Duntsch, I., Grzymała-Busse, J., Orłowska, E., Polkowski, L. (eds.) Transactions on Rough Sets VI. LNCS, vol. 4374, pp. 1–25. Springer, Heidelberg (2007). https://​doi.​org/​10.​1007/​978-3-540-71200-8_​1
  7. Bezhanishvili, G.: Varieties of monadic Heyting algebras. part I. Studia Logica 61(3), 367–402 (1998)
  8. Bezhanishvili, N., de Jongh D.: Intuitionistic logic. Institute for Logic, Language and Computation (ILLC), University of Amsterdam PP-2006-25 (2006)
  9. Birkhoff, G.: Lattice Theory. Colloquium Publications, vol. XXV, 3rd edn., American Mathematical Society, Providence (1995)
  10. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press (1991)
  11. Boicescu, V., Filipoiu, A., Georgescu, G., Rudeanu, S.: Łukasiewicz-Moisil Algebras. North-Holland, Amsterdam (1991)MATH
  12. Bonikowski, Z.: Algebraic structures of rough sets in representative approximation spaces. Electronic Notes Theoret. Comput. Sci. 82, 1–12 (2003)MATHView Article
  13. Bonikowski, Z., Bryniariski, E., Skardowska, V.W.: Extension and intension in the rough set theory. Inf. Sci. 107, 149–167 (1998)MATHView Article
  14. Chakraborty, M.K., Banerjee, M.: Rough sets: some foundational issues. Fundamenta Informaticae 127, 1–15 (2013)MathSciNetMATHView Article
  15. Cignoli, R.: Boolean elements in Łukasiewicz algebras. I. Proc. Japan Acad. 41, 670–675 (1965)MathSciNetMATHView Article
  16. Cignoli, R.: The class of Kleene algebras satisfying an interpolation property and Nelson algebras. Algebra Universalis 23(3), 262–292 (1986)MathSciNetMATHView Article
  17. Cignoli, R.: The algebras of Łukasiewicz many-valued logic: a historical overview. In: Aguzzoli, S., Ciabattoni, A., Gerla, B., Manara, C., Marra, V. (eds.) Algebraic and Proof-theoretic Aspects of Non-classical Logics. LNAI, vol. 4460, pp. 69–83. Springer, Heidelberg (2007). https://​doi.​org/​10.​1007/​978-3-540-75939-3_​5View ArticleMATH
  18. Ciucci, D., Dubois, D.: Three-valued logics, uncertainty management and rough sets. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets XVII. LNCS, vol. 8375, pp. 1–32. Springer, Heidelberg (2014). https://​doi.​org/​10.​1007/​978-3-642-54756-0_​1View ArticleMATH
  19. Comer, S.: Perfect extensions of regular double Stone algebras. Algebra Universalis 34(1), 96–109 (1995)MathSciNetMATHView Article
  20. Dai, J.-H.: Logic for rough sets with rough double stone algebraic semantics. In: Slezak, D., Wang, G., Szczuka, M., Düntsch, I., Yao, Y. (eds.) RSFDGrC 2005. LNCS (LNAI), vol. 3641, pp. 141–148. Springer, Heidelberg (2005). https://​doi.​org/​10.​1007/​11548669_​15View Article
  21. Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order. Cambridge University Press (2002)
  22. Degang, C., Wenxiu, Z., Yeung, D., Tsang, E.: Rough approximations on a complete completely distributive lattice with applications to generalized rough sets. Inf. Sci. 176, 1829–1848 (2006)MathSciNetMATHView Article
  23. Dunn, J.: The algebra of intensional logic. Doctoral dissertation, University of Pittsburgh (1966)
  24. Dunn, J.: Star and perp: two treatments of negation. In: Tomberlin, J. (ed.) Philosophical Perspectives, vol. 7, pp. 331–357. Ridgeview Publishing Company, Atascadero, California (1994)
  25. Dunn, J.: Positive modal logic. Studia Logica 55, 301–317 (1995)MathSciNetMATHView Article
  26. Dunn, J.: Generalised ortho negation. In: Wansing, H. (ed.) Negation: A Notion in Focus, pp. 3–26. Walter de Gruyter, Berlin (1996)
  27. Dunn, J.: A comparative study of various model-theoretic treatments of negation: a history of formal negations. In: Gabbay, D., Wansing, H. (eds.) What is Negation?, pp. 23–51. Kluwer Academic Publishers, Netherlands (1999)View Article
  28. Dunn, J.: Partiality and its dual. Studia Logica 66, 5–40 (2000)MathSciNetMATHView Article
  29. Dunn, J.: Negation in the context of gaggle theory. Studia Logica 80, 235–264 (2005)MathSciNetMATHView Article
  30. Düntsch, I.: A logic for rough sets. Theoret. Comput. Sci. 179, 427–436 (1997)MathSciNetMATHView Article
  31. Düntsch, I., Orłowska, E.: Discrete dualities for double Stone algebras. Studia Logica 99(1), 127–142 (2011)MathSciNetMATHView Article
  32. Fidel, M.: An algebraic study of a propositional system of Nelson. In: Arruda, A.I., da Costa, N.C.A., Chuaqui, R. (eds.) Mathematical Logic: Proceedings of First Brazilian Conference, pp. 99–117. Lecture Notes in Pure and Applied Mathematics, vol. 39, M.Dekker Inc., New York (1978)
  33. Gehrke, M., van Gool, S.J.: Distributive envelopes and topological duality for lattices via canonical extensions. Order 31(3), 435–461 (2014)MathSciNetMATHView Article
  34. Gehrke, M., Walker, E.: On the structure of rough sets. Bull. Polish Acad. Sci. Math. 40(3), 235–255 (1992)
  35. Greco, S., Matarazzo, B., Słowinski, R.: Rough sets theory for multi-criteria decision analysis. Eur. J. Oper. Res. 129, 1–47 (2001)MATHView Article
  36. Greco, S., Matarazzo, B., Slowinski, R.: Multicriteria classification by dominance based rough set approach. In: Kloesgen, W., Zytkow, J. (eds.) Handbook of Data Mining and Knowledge discovery. Oxford University Press, New York (2002)MATH
  37. Greco, S., Matarazzo, B., Słowinski, R.: Algebra and topology for dominance-based rough set approach. In: Ras, Z., Tsay, L.S. (eds.) Advances in Intelligent Information Systems, pp. 43–78. Springer, Heidelberg (2010). https://​doi.​org/​10.​1007/​978-3-642-05183-8_​3View Article
  38. Iturrioz, L.: Rough sets and three-valued structures. In: Orłowska, E. (ed.) Logic at Work: Essays Dedicated to the Memory of Helena Rasiowa. Studies in Fuzziness and Soft Computing, vol. 24, pp. 596–603. Springer, Heidelberg (1999)
  39. Järvinen, J., Pagliani, P., Radeleczki, S.: Information completeness in Nelson algebras of rough sets induced by quasiorders. Studia Logica 101(5), 1073–1092 (2013)MathSciNetMATHView Article
  40. Järvinen, J., Radeleczki, S.: Representation of Nelson algebra by rough sets determined by quasiorder. Algebra Universalis 66, 163–179 (2011)MathSciNetMATHView Article
  41. Järvinen, J., Radeleczki, S.: Rough sets determined by tolerances. Int. J. Approximate Reason. 55(6), 1419–1438 (2014)MathSciNetMATHView Article
  42. Järvinen, J., Radeleczki, S., Veres, L.: Rough sets determined by quasiorders. Order 26, 337–355 (2009)MathSciNetMATHView Article
  43. Kalman, J.: Lattices with involution. Trans. Am. Math. Soc. 87, 485–491 (1958)MathSciNetMATHView Article
  44. Katriňák, T.: Construction of regular double p-algebras. Bull. Soc. Roy. Sci. Liege 43, 238–246 (1974)MathSciNetMATH
  45. Khan, M.A., Banerjee, M.: Logics for information systems and their dynamic extensions. ACM Trans. Comput. Logic 12(4), art. no. 29 (2011)
  46. Khan, M.A., Banerjee, M., Rieke, R.: An update logic for information systems. Int. J. Approximate Reasoning 55(1), 436–456 (2014)MathSciNetMATHView Article
  47. Kozen, D.: On kleene algebras and closed semirings. In: Rovan, B. (ed.) MFCS 1990. LNCS, vol. 452, pp. 26–47. Springer, Heidelberg (1990). https://​doi.​org/​10.​1007/​BFb0029594View Article
  48. Kumar, A., Banerjee, M.: Definable and rough sets in covering-based approximation spaces. In: Li, T., Nguyen, H., Wang, G., Grzymała-Busse, J., Janicki, R., Hassanien, A., Yu, H. (eds.) Rough Sets and Knowledge Technology, pp. 488–495. Springer, Heidelberg (2012). https://​doi.​org/​10.​1007/​978-3-642-31900-6_​60View Article
  49. Kumar, A., Banerjee, M.: Algebras of definable and rough sets in quasi order-based approximation spaces. Fundamenta Informaticae 141, 37–55 (2015)MathSciNetMATHView Article
  50. Kumar, A., Banerjee, M.: Kleene algebras and logic: boolean and rough set representations, 3-valued, rough set and perp semantics. Studia Logica 105, 439–469 (2017)MathSciNetMATHView Article
  51. Kumar, A., Banerjee, M.: A semantic analysis of Stone and dual Stone negations with regularity. In: Ghosh, S., Prasad, S. (eds.) ICLA 2017, pp. 139–153. Springer, Heidelberg (2017). https://​doi.​org/​10.​1007/​978-3-662-54069-5_​11View Article
  52. Li, T.J.: Rough approximation operators in covering approximation spaces. In: Greco, S., et al. (eds.) RSCTC 2006, pp. 174–182. Springer, Heidelberg (2006). https://​doi.​org/​10.​1007/​11908029_​20View Article
  53. Nagarajan, E.K.R., Umadevi, D.: A method of representing rough sets system determined by quasi orders. Order 30, 313–337 (2013)MathSciNetMATHView Article
  54. Ono, H.: On some intuitionistic modal logic. Publ. Inst. Math. Sci. Kyoto Univ. 13, 687–722 (1977)MathSciNetMATHView Article
  55. Orłowska, E., Rewitzky, I.: Duality via truth: semantic frameworks for lattice-based logics. Logic J. IGPL 13(4), 467–490 (2005)MathSciNetMATHView Article
  56. Orłowska, E., Rewitzky, I.: Discrete duality for Heyting algebras with operators. Fundamenta Informaticae 81(1–3), 275–295 (2007)MathSciNetMATH
  57. Orłowska, E., Rewitzky, I., Düntsch, I.: Relational semantics through duality. In: MacCaull, W., Winter, M., Düntsch, I. (eds.) Relational Methods in Computer Science, pp. 17–32. Springer, Heidelberg (2006). https://​doi.​org/​10.​1007/​11734673_​2View ArticleMATH
  58. Pagliani, P.: Rough sets and Nelson algebras. Fundamenta Informaticae 27(2–3), 205–219 (1996)MathSciNetMATHView Article
  59. Pagliani, P.: Rough set theory and logic-algebraic structures. In: Orłowska, E. (ed.) Incomplete Information: Rough Set Analysis. Studies in Fuzziness and Soft Computing, vol. 13, pp. 109–190. Springer, Heidelberg (1998). https://​doi.​org/​10.​1007/​978-3-7908-1888-8_​6
  60. Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11, 341–356 (1982)MATHView Article
  61. Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning About Data. Kluwer Academic Publishers (1991)
  62. Pomykała, J., Pomykała, J.A.: The Stone algebra of rough sets. Bull. Polish Acad. Sci. Math. 36, 495–508 (1988)
  63. Pomykała, J.A.: Approximation, similarity and rough construction. ILLC prepublication series CT-93-07, University of Amsterdam (1993)
  64. Qin, K., Gao, Y., Pei, Z.: On covering rough sets. In: Yao, J.T., Lingras, P., Wu, W.-Z., Szczuka, M., Cercone, N.J., Śezak, D. (eds.) RSKT 2007. LNCS (LNAI), vol. 4481, pp. 34–41. Springer, Heidelberg (2007). https://​doi.​org/​10.​1007/​978-3-540-72458-2_​4View Article
  65. Rasiowa, H.: An Algebraic Approach to Non-classical Logics. North-Holland (1974)
  66. Restall, G.: Defining double negation elimination. L. J. IGPL 8(6), 853–860 (2000)MathSciNetMATHView Article
  67. Saha, A., Sen, J., Chakraborty, M.K.: Algebraic structures in the vicinity of pre-rough algebra and their logics. Inf. Sci. 282, 296–320 (2014)MathSciNetMATHView Article
  68. Saha, A., Sen, J., Chakraborty, M.K.: Algebraic structures in the vicinity of pre-rough algebra and their logics II. Inf. Sci. 333, 44–60 (2016)MathSciNetMATHView Article
  69. Samanta, P., Chakraborty, M.K.: Generalized rough sets and implication lattices. In: Peters, J.F., et al. (eds.) Transactions on Rough Sets XIV. LNCS, vol. 6600, pp. 183–201. Springer, Heidelberg (2011). https://​doi.​org/​10.​1007/​978-3-642-21563-6_​10View ArticleMATH
  70. Samanta, P., Chakraborty, M.K.: Interface of rough set systems and modal logics: a survey. In: Peters, J.F., Skowron, A., Ślȩzak, D., Nguyen, H.S., Bazan, J.G. (eds.) Transactions on Rough Sets XIX. LNCS, vol. 8988, pp. 114–137. Springer, Heidelberg (2015). https://​doi.​org/​10.​1007/​978-3-662-47815-8_​8View Article
  71. Sanjuan, E.: Heyting algebras with Boolean operators of rough sets and information retrieval applications. Discrete Appl. Math. 156, 967–983 (2008)MathSciNetMATHView Article
  72. Suzuki, N.: An algebraic approach to intuitionistic modal logics in connection with intermediate predicate logics. Studia Logica 48(2), 141–155 (1989)MathSciNetMATHView Article
  73. Urquhart, A.: Basic many-valued logic. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 2, pp. 249–295. Springer, Heidelberg (2001). https://​doi.​org/​10.​1007/​978-94-017-0452-6_​4View ArticleMATH
  74. Vakarelov, D.: Notes on N-lattices and constructive logic with strong negation. Studia Logica 36, 109–125 (1977)MathSciNetMATHView Article
  75. Varlet, J.: A regular variety of type (2,2,1,1,0,0). Algebra Universalis 2, 218–223 (1972)MathSciNetMATHView Article
  76. Yao, Y., Chen, Y.: Rough set approximations in formal concept analysis. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets V. LNCS, vol. 4100, pp. 285–305. Springer, Heidelberg (2006). https://​doi.​org/​10.​1007/​11847465_​14View Article
  77. Yao, Y.: Relational interpretations of neighborhood operators and rough set approximation operators. Inf. Sci. 111, 239–259 (1998)MathSciNetMATHView Article
  78. Yao, Y.: Granular computing using neighborhood system. In: Roy, R., Furuhashi, T., Chawdhry, P.K. (eds.) Advances in Soft Computing: Engineering Design and Manufacturing, pp. 539–553. Springer, New York (1999). https://​doi.​org/​10.​1007/​978-1-4471-0819-1_​40View Article
  79. Yao, Y.: Rough sets, neighborhood system and granular computing. In: Proceedings of IEEE Canadian Conference on Electrical and Computer Engineering, pp. 1553–1558. IEEE Press (1999)
  80. Yao, Y.: Information granulation and rough set approximation. Int. J. Intell. Syst. 16(1), 87–104 (2001)MathSciNetMATHView Article
  81. Yao, Y.: Covering based rough set approximations. Inf. Sci. 200, 91–107 (2012)MathSciNetMATHView Article
  82. Yao, Y., Lin, T.: Generalization of rough sets using modal logic. Intell. Autom. Soft Comput. Int. J. 2(2), 103–120 (1996)View Article
  83. Zhang, Y., Luo, M.: Relationships between covering-based rough sets and relation based rough sets. Inf. Sci. 225, 55–71 (2013)MathSciNetMATHView Article
  84. Zhoua, N., Hua, B.: Rough sets based on complete completely distributive lattice. Inf. Sci. 269, 378–387 (2014)MathSciNetView Article
  85. Zhu, W.: Topological approaches to covering rough sets. Inf. Sci. 177, 1499–1508 (2007)MathSciNetMATHView Article
  86. Zhu, W., Wang, F.Y.: Reduction and axiomatization of covering generalized rough sets. Inf. Sci. 152, 217–230 (2003)MATHView Article
  87. Zhu, W., Wang, F.Y.: Relationship among three types of covering rough sets. In: Proceedings of IEEE International Conference on Granular Computing, pp. 43–48 (2006)
Metadata
Title
A Study of Algebras and Logics of Rough Sets Based on Classical and Generalized Approximation Spaces
Author
Arun Kumar
Copyright Year
2020
Publisher
Springer Berlin Heidelberg
Book
Transactions on Rough Sets XXII

Print ISBN: 978-3-662-62797-6

Electronic ISBN: 978-3-662-62798-3

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-3-662-62798-3_4

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