Algebraic Semantics of Proto-Transitive Rough Sets

Part of the Lecture Notes in Computer Science book series (LNCS, volume 10020)


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.


Proto-transitive relations Generalized transitivity Rough dependence Rough objects Granulation Algebraic semantics Approximate relations Approximate semantics Nelson algebras Axiomatic theory of granules Contamination problem Knowledge 



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.


  1. 1.
    Bianucci, D., Cattaneo, G., Ciucci, D.: Entropies and co-entropies of coverings with application to incomplete information systems. Fundam. Inform. 75, 77–105 (2007)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Burmeister, P.: A Model-Theoretic Oriented Approach to Partial Algebras. Akademie-Verlag, Berlin (1986, 2002)Google Scholar
  3. 3.
    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)CrossRefGoogle Scholar
  4. 4.
    Chajda, I., Haviar, M.: Induced pseudo orders. Acta Univ. Palack. Olomou 30(1), 9–16 (1991)MathSciNetzbMATHGoogle Scholar
  5. 5.
    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)CrossRefGoogle Scholar
  6. 6.
    Ciucci, D.: Approximation algebra and framework. Fundam. Inform. 94, 147–161 (2009)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Dimitrov, B.: Some Obreshkov measures of dependence and their Use. Comptes Rendus Acad Bulg. Sci. 63(1), 5–18 (2010)MathSciNetzbMATHGoogle Scholar
  8. 8.
    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)CrossRefGoogle Scholar
  9. 9.
    Greco, S., Pawlak, Z., Slowinski, R.: Can Bayesian measures be useful for rough set decision making? Eng. Appl. AI 17, 345–361 (2004)CrossRefGoogle Scholar
  10. 10.
    Hajek, A.: Fifteen arguments against hypothetical frequentism. Erkenntnis 211(70), 211–235 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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)CrossRefGoogle Scholar
  12. 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)CrossRefGoogle Scholar
  13. 13.
    Jarvinen, J., Pagliani, P., Radeleczki, S.: Information completeness in Nelson algebras of rough sets induced by quasiorders. Stud. Logica. 101, 1–20 (2012)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Jarvinen, J., Radeleczki, S.: Representation of Nelson algebras by rough sets determined by quasi-orders. Algebra Univers. 66, 163–179 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Keet, C.M.: A formal theory of granules - Phd thesis. Ph.D. thesis, Faculty of Computer Science, Free University of Bozen (2008)Google Scholar
  16. 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)CrossRefGoogle Scholar
  17. 17.
    Ljapin, E.S.: Partial Algebras and Their Applications. Kluwer Academic, Dordrecht (1996)Google Scholar
  18. 18.
    Makinson, D.: General Patterns in Nonmonotonic Reasoning, vol. 3, pp. 35–110. Oxford University Press, New York (1994)Google Scholar
  19. 19.
    Makinson, D.: Bridges between classical and nonmonotonic logic. Logic J. IGPL 11, 69–96 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mani, A.: Algebraic semantics of similarity-based bitten rough set theory. Fundam. Inform. 97(1–2), 177–197 (2009)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Mani, A.: Meaning, choice and similarity based rough set theory. In: International Conference Logic and Application, January 2009, Chennai, pp. 1–12 (2009).
  22. 22.
    Mani, A.: Choice inclusive general rough semantics. Inf. Sci. 181(6), 1097–1115 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  23. 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)CrossRefGoogle Scholar
  24. 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)CrossRefGoogle Scholar
  25. 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)Google Scholar
  26. 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).
  27. 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)CrossRefGoogle Scholar
  28. 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)CrossRefGoogle Scholar
  29. 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. IntellMathSciNetGoogle Scholar
  30. 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 CrossRefGoogle Scholar
  31. 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).
  32. 32.
    Mani, A.: Probabilities, dependence and rough membership functions. Int. J. Comput. Appl. (Special Issue on Comput. Intell.), 1–27 (2016, accepted)Google Scholar
  33. 33.
    Moore, E.F., Shannon, C.E.: Reliable circuits using less reliable relays-I, II. Bell Syst. Tech. J. 191–208, 281–297 (1956)Google Scholar
  34. 34.
    Pagliani, P., Chakraborty, M.: A Geometry of Approximation: Rough Set Theory: Logic, Algebra and Topology of Conceptual Patterns. Springer, Berlin (2008)zbMATHGoogle Scholar
  35. 35.
    Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning About Data. Kluwer Academic Publishers, Dodrecht (1991)CrossRefzbMATHGoogle Scholar
  36. 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 2002Google Scholar
  37. 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)CrossRefGoogle Scholar
  38. 38.
    Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27(379–423), 623–656 (1948)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 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)CrossRefGoogle Scholar
  40. 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)CrossRefGoogle Scholar
  41. 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)CrossRefGoogle Scholar
  42. 42.
    Yao, Y.: Information granulation and rough set approximation. Int. J. Intell. Syst. 16, 87–104 (2001)CrossRefzbMATHGoogle Scholar
  43. 43.
    Yao, Y.: Probabilistic approach to rough sets. Expert Syst. 20(5), 287–297 (2003)CrossRefGoogle Scholar
  44. 44.
    Yao, Y.: Probabilistic rough set approximations. Int. J. Approximate Reasoning 49, 255–271 (2008)CrossRefzbMATHGoogle Scholar
  45. 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)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2016

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of CalcuttaKolkataIndia

Personalised recommendations