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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References
Bianucci, D., Cattaneo, G., Ciucci, D.: Entropies and co-entropies of coverings with application to incomplete information systems. Fundam. Inform. 75, 77–105 (2007)
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)
Chajda, I., Haviar, M.: Induced pseudo orders. Acta Univ. Palack. Olomou 30(1), 9–16 (1991)
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)
Ciucci, D.: Approximation algebra and framework. Fundam. Inform. 94, 147–161 (2009)
Dimitrov, B.: Some Obreshkov measures of dependence and their Use. Comptes Rendus Acad Bulg. Sci. 63(1), 5–18 (2010)
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)
Greco, S., Pawlak, Z., Slowinski, R.: Can Bayesian measures be useful for rough set decision making? Eng. Appl. AI 17, 345–361 (2004)
Hajek, A.: Fifteen arguments against hypothetical frequentism. Erkenntnis 211(70), 211–235 (2009)
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)
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)
Jarvinen, J., Pagliani, P., Radeleczki, S.: Information completeness in Nelson algebras of rough sets induced by quasiorders. Stud. Logica. 101, 1–20 (2012)
Jarvinen, J., Radeleczki, S.: Representation of Nelson algebras by rough sets determined by quasi-orders. Algebra Univers. 66, 163–179 (2011)
Keet, C.M.: A formal theory of granules - Phd thesis. Ph.D. thesis, Faculty of Computer Science, Free University of Bozen (2008)
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)
Ljapin, E.S.: Partial Algebras and Their Applications. Kluwer Academic, Dordrecht (1996)
Makinson, D.: General Patterns in Nonmonotonic Reasoning, vol. 3, pp. 35–110. Oxford University Press, New York (1994)
Makinson, D.: Bridges between classical and nonmonotonic logic. Logic J. IGPL 11, 69–96 (2003)
Mani, A.: Algebraic semantics of similarity-based bitten rough set theory. Fundam. Inform. 97(1–2), 177–197 (2009)
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
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
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)
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)
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)
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
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)
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)
Mani, A.: Ontology, rough Y-systems and dependence. Int. J. Comput. Sci. Appl. 11(2), 114–136 (2014). Special Issue of IJCSA on Comput. Intell
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
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
Mani, A.: Probabilities, dependence and rough membership functions. Int. J. Comput. Appl. (Special Issue on Comput. Intell.), 1–27 (2016, accepted)
Moore, E.F., Shannon, C.E.: Reliable circuits using less reliable relays-I, II. Bell Syst. Tech. J. 191–208, 281–297 (1956)
Pagliani, P., Chakraborty, M.: A Geometry of Approximation: Rough Set Theory: Logic, Algebra and Topology of Conceptual Patterns. Springer, Berlin (2008)
Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning About Data. Kluwer Academic Publishers, Dodrecht (1991)
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
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)
Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27(379–423), 623–656 (1948)
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)
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)
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)
Yao, Y.: Information granulation and rough set approximation. Int. J. Intell. Syst. 16, 87–104 (2001)
Yao, Y.: Probabilistic approach to rough sets. Expert Syst. 20(5), 287–297 (2003)
Yao, Y.: Probabilistic rough set approximations. Int. J. Approximate Reasoning 49, 255–271 (2008)
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)
Acknowledgment
The present author would like to thank Prof Mihir Chakraborty for discussions on PRAX and the referee for drawing attention to [8] and related references in particular.
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Mani, A. (2016). Algebraic Semantics of Proto-Transitive Rough Sets. In: Peters, J., Skowron, A. (eds) Transactions on Rough Sets XX. Lecture Notes in Computer Science(), vol 10020. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53611-7_3
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