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2016 | OriginalPaper | Chapter

Algebraic Semantics of Proto-Transitive Rough Sets

Author : A. Mani

Published in: Transactions on Rough Sets XX

Publisher: Springer Berlin Heidelberg

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Abstract

Rough Sets over generalized transitive relations like proto-transitive ones have been initiated recently by the present author. In a recent paper, approximation of proto-transitive relations by other relations was investigated and the relation with rough approximations was developed towards constructing semantics that can handle fragments of structure. It was also proved that difference of approximations induced by some approximate relations need not induce rough structures. In this research, the structure of rough objects is characterized and a theory of dependence for general rough sets is developed and used to internalize the Nelson-algebra based approximate semantics developed earlier by the present author. This is part of the different semantics of PRAX developed in this paper by her. The theory of rough dependence initiated in earlier papers is extended in the process. This paper is reasonably self-contained and includes proofs and extensions of representation of objects that have not been published earlier.

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Bianucci, D., Cattaneo, G., Ciucci, D.: Entropies and co-entropies of coverings with application to incomplete information systems. Fundam. Inform. 75, 77–105 (2007)MathSciNetMATH

Burmeister, P.: A Model-Theoretic Oriented Approach to Partial Algebras. Akademie-Verlag, Berlin (1986, 2002)

Cattaneo, G., Ciucci, D.: Lattices with interior and closure operators and abstract approximation spaces. In: Peters, J.F., Skowron, A., Wolski, M., Chakraborty, M.K., Wu, W.-Z. (eds.) Transactions on Rough Sets X. LNCS, vol. 5656, pp. 67–116. Springer, Heidelberg (2009)CrossRef

Chajda, I., Haviar, M.: Induced pseudo orders. Acta Univ. Palack. Olomou 30(1), 9–16 (1991)MathSciNetMATH

Chakraborty, M.K., Samanta, P.: Consistency-degree between knowledges. In: Kryszkiewicz, M., Peters, J.F., Rybiński, H., Skowron, A. (eds.) RSEISP 2007. LNCS (LNAI), vol. 4585, pp. 133–141. Springer, Heidelberg (2007)CrossRef

Ciucci, D.: Approximation algebra and framework. Fundam. Inform. 94, 147–161 (2009)MathSciNetMATH

Dimitrov, B.: Some Obreshkov measures of dependence and their Use. Comptes Rendus Acad Bulg. Sci. 63(1), 5–18 (2010)MathSciNetMATH

Ganter, B., Meschke, C.: A formal concept analysis approach to rough data tables. In: Peters, J.F., Skowron, A., Sakai, H., Chakraborty, M.K., Slezak, D., Hassanien, A.E., Zhu, W. (eds.) Transactions on Rough Sets XIV. LNCS, vol. 6600, pp. 37–61. Springer, Heidelberg (2011)CrossRef

Greco, S., Pawlak, Z., Slowinski, R.: Can Bayesian measures be useful for rough set decision making? Eng. Appl. AI 17, 345–361 (2004)CrossRef

10.

Hajek, A.: Fifteen arguments against hypothetical frequentism. Erkenntnis 211(70), 211–235 (2009)MathSciNetCrossRefMATH

11.

Janicki, R.: Approximations of arbitrary binary relations by partial orders: classical and rough set models. In: Peters, J.F., Skowron, A., Chan, C.-C., Grzymala-Busse, J.W., Ziarko, W.P. (eds.) Transactions on Rough Sets XIII. LNCS, vol. 6499, pp. 17–38. Springer, Heidelberg (2011)CrossRef

12.

Järvinen, J.: Lattice theory for rough sets. In: Peters, J.F., Skowron, A., Düntsch, I., Grzymała-Busse, J.W., Orłowska, E., Polkowski, L. (eds.) Transactions on Rough Sets VI. LNCS, vol. 4374, pp. 400–498. Springer, Heidelberg (2007)CrossRef

13.

Jarvinen, J., Pagliani, P., Radeleczki, S.: Information completeness in Nelson algebras of rough sets induced by quasiorders. Stud. Logica. 101, 1–20 (2012)MathSciNetMATH

14.

Jarvinen, J., Radeleczki, S.: Representation of Nelson algebras by rough sets determined by quasi-orders. Algebra Univers. 66, 163–179 (2011)MathSciNetCrossRefMATH

15.

Keet, C.M.: A formal theory of granules - Phd thesis. Ph.D. thesis, Faculty of Computer Science, Free University of Bozen (2008)

16.

Lin, T.Y.: Granular computing -1: the concept of Granulation and its formal model. Int. J. Granular Comput. Rough Sets Int. Syst. 1(1), 21–42 (2009)CrossRef

17.

Ljapin, E.S.: Partial Algebras and Their Applications. Kluwer Academic, Dordrecht (1996)

18.

Makinson, D.: General Patterns in Nonmonotonic Reasoning, vol. 3, pp. 35–110. Oxford University Press, New York (1994)

19.

Makinson, D.: Bridges between classical and nonmonotonic logic. Logic J. IGPL 11, 69–96 (2003)MathSciNetCrossRefMATH

20.

Mani, A.: Algebraic semantics of similarity-based bitten rough set theory. Fundam. Inform. 97(1–2), 177–197 (2009)MathSciNetMATH

21.

Mani, A.: Meaning, choice and similarity based rough set theory. In: International Conference Logic and Application, January 2009, Chennai, pp. 1–12 (2009). http://arxiv.org/abs/0905.1352

22.

Mani, A.: Choice inclusive general rough semantics. Inf. Sci. 181(6), 1097–1115 (2011). http://dx.doi.org/10.1016/j.ins.2010.11.016 MathSciNetCrossRefMATH

23.

Mani, A.: Axiomatic granular approach to knowledge correspondences. In: Li, T., Nguyen, H.S., Wang, G., Grzymala-Busse, J., Janicki, R., Hassanien, A.E., Yu, H. (eds.) RSKT 2012. LNCS, vol. 7414, pp. 482–487. Springer, Heidelberg (2012)CrossRef

24.

Mani, A.: Dialectics of counting and the mathematics of vagueness. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets XV. LNCS, vol. 7255, pp. 122–180. Springer, Heidelberg (2012)CrossRef

25.

Mani, A.: Contamination-free measures and Algebraic operations. In: Pal, N., et al. (ed.) Fuzzy Systems (FUZZ), 2013 IEEE International Conference on Fuzzy Systems, vol. F-1438, Hyderabad, India, pp. 1–8 (2013)

26.

Mani, A.: Dialectics of knowledge representation in a granular rough set theory. In: Refereed Conference Paper: ICLA 2013, Institute of Mathematical Sciences, Chennai, pp. 1–12 (2013). http://arxiv.org/abs/1212.6519

27.

Mani, A.: Towards logics of some rough perspectives of knowledge. In: Skowron, A., Suraj, Z. (eds.) Rough Sets and Intelligent Systems - Professor Zdzisław Pawlak in Memoriam. ISRL, vol. 43, pp. 419–444. Springer, Heidelberg (2013)CrossRef

28.

Mani, A.: Approximation dialectics of proto-transitive rough sets. In: Chakraborty, M.K., Skowron, A., Maiti, M., Kar, A. (eds.) Facets of Uncertainties and Applications. Springer Proceedings in Mathematics and Statistics, vol. 125. Springer, New Delhi (2015)CrossRef

29.

Mani, A.: Ontology, rough Y-systems and dependence. Int. J. Comput. Sci. Appl. 11(2), 114–136 (2014). Special Issue of IJCSA on Comput. IntellMathSciNet

30.

Mani, A.: Antichain based semantics for rough sets. In: Ciucci, D., et al. (eds.) RSKT 2015. LNCS, vol. 9436, pp. 335–346. Springer, Heidelberg (2015). doi:10.1007/978-3-319-25754-9_30 CrossRef

31.

Mani, A.: Types of Probabilities Associated with Rough Membership Functions. In: Bhattacharyya, S., et al. (ed.) Proceedings of ICRCICN 2015: IEEE Xplore, pp. 175–180. IEEE Computer Society (2015). http://dx.doi.org/10.1109/ICRCICN.2015.7434231

32.

Mani, A.: Probabilities, dependence and rough membership functions. Int. J. Comput. Appl. (Special Issue on Comput. Intell.), 1–27 (2016, accepted)

33.

Moore, E.F., Shannon, C.E.: Reliable circuits using less reliable relays-I, II. Bell Syst. Tech. J. 191–208, 281–297 (1956)

34.

Pagliani, P., Chakraborty, M.: A Geometry of Approximation: Rough Set Theory: Logic, Algebra and Topology of Conceptual Patterns. Springer, Berlin (2008)MATH

35.

Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning About Data. Kluwer Academic Publishers, Dodrecht (1991)CrossRefMATH

36.

Pawlak, Z.: Decision tables and decision spaces. In: Proceedings of the 6th International Conference on Soft Computing and Distributed Processing (SCDP 2002), 24–25 June 2002

37.

Pawlak, Z.: Some issues on rough sets. In: Peters, J.F., Skowron, A., Grzymała-Busse, J.W., Kostek, B., Świniarski, R.W., Szczuka, M.S., et al. (eds.) Transactions on Rough Sets I. LNCS, vol. 3100, pp. 1–58. Springer, Heidelberg (2004)CrossRef

38.

Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27(379–423), 623–656 (1948)MathSciNetCrossRefMATH

39.

Ślęzak, D.: Rough sets and bayes factor. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets III. LNCS, vol. 3400, pp. 202–229. Springer, Heidelberg (2005)CrossRef

40.

Ślȩzak, D., Wasilewski, P.: Granular sets – foundations and case study of tolerance spaces. In: An, A., Stefanowski, J., Ramanna, S., Butz, C.J., Pedrycz, W., Wang, G. (eds.) RSFDGrC 2007. LNCS (LNAI), vol. 4482, pp. 435–442. Springer, Heidelberg (2007)CrossRef

41.

Wasilewski, P., Slezak, D.: Foundations of rough sets from vagueness perspective. In: Hassanien, A., et al. (eds.) Rough Computing: Theories, Technologies and Applications, pp. 1–37. Information Science Reference, IGI, Global, Hershey (2008)CrossRef

42.

Yao, Y.: Information granulation and rough set approximation. Int. J. Intell. Syst. 16, 87–104 (2001)CrossRefMATH

43.

Yao, Y.: Probabilistic approach to rough sets. Expert Syst. 20(5), 287–297 (2003)CrossRef

44.

Yao, Y.: Probabilistic rough set approximations. Int. J. Approximate Reasoning 49, 255–271 (2008)CrossRefMATH

45.

Zadeh, L.A.: Fuzzy sets and information granularity. In: Gupta, N., et al. (eds.) Advances in Fuzzy Set Theory and Applications, pp. 3–18. North Holland, Amsterdam (1979)

Title: Algebraic Semantics of Proto-Transitive Rough Sets
Author: A. Mani
Publisher: Springer Berlin Heidelberg
Book: Transactions on Rough Sets XX
Print ISBN: 978-3-662-53610-0

Electronic ISBN: 978-3-662-53611-7

Copyright Year: 2016
DOI: https://doi.org/10.1007/978-3-662-53611-7_3

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