Skip to main content
SpringerLink
Log in

A Logic for Multiple-source Approximation Systems with Distributed Knowledge Base

  • Published:
Journal of Philosophical Logic Aims and scope Submit manuscript

Abstract

The theory of rough sets starts with the notion of an approximation space, which is a pair (U,R), U being the domain of discourse, and R an equivalence relation on U. R is taken to represent the knowledge base of an agent, and the induced partition reflects a granularity of U that is the result of a lack of complete information about the objects in U. The focus then is on approximations of concepts on the domain, in the context of the granularity. The present article studies the theory in the situation where information is obtained from different sources. The notion of approximation space is extended to define a multiple-source approximation system with distributed knowledge base, which is a tuple \((U,R_P)_{P\ss_f N}\), where N is a set of sources and P ranges over all finite subsets of N. Each R P is an equivalence relation on U satisfying some additional conditions, representing the knowledge base of the group P of sources. Thus each finite group of sources and hence individual source perceives the same domain differently (depending on what information the group/individual source has about the domain), and the same concept may then have approximations that differ with the groups. In order to express the notions and properties related with rough set theory in this multiple-source situation, a quantified modal logic LMSAS D is proposed. In LMSAS D, quantification ranges over modalities, making it different from modal predicate logic and modal logic with propositional quantifiers. Some fragments of LMSAS D are discussed and it is shown that the modal system KTB is embedded in LMSAS D. The epistemic logic \(S5^D_n\) is also embedded in LMSAS D, and cannot replace the latter to serve our purpose. The relationship of LMSAS D with first and second-order logics is presented. Issues of expressibility, axiomatization and decidability are addressed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Balbiani, P. (1998). Axiomatization of logics based on Kripke models with relative accessibility relations. In E. Orłowska (Ed.), Incomplete information: Rough set analysis (pp. 553–578). Heidelberg: Physica-Verlag.

    Google Scholar 

  2. Balbiani, P., & Orłowska, E. (1999). A hierarchy of modal logics with relative accessibility relations. Journal of Applied Non-Classical Logics, 9(2–3), 303–328.

    Google Scholar 

  3. Banerjee, M., & Khan, M. A. (2007). Propositional logics from rough set theory. Transactions on Rough Sets, 6, 1–25.

    Google Scholar 

  4. Blackburn, P., de Rijke, M., & Venema, Y. (2001). Modal logic. Cambridge University Press.

  5. Chang, C. C., & Keisler, H. J. (1992). Model theory. Elsevier Science Publications B.V.

  6. Chellas, B. F. (1980). Modal logic. Cambridge University Press.

  7. Demri, S., & Orłowska, E. (2002). Incomplete information: Structure, inference, complexity. Springer.

  8. Fagin, R., Halpern, J. Y., Moses, Y., & Vardi, M. Y. (1995). Reasoning about knowledge. The MIT.

  9. Farinas Del Cerro, L., & Orłowska, E. (1997). DAL—a logic for data analysis. Theoretical Computer Science, 36, 251–264.

    Article  Google Scholar 

  10. Greco, S., Matarazzo, B., & Słowiński, R. (2008). Parametrized rough set model using rough membership and Bayesian confirmation measures. International Journal of Approximate Reasoning, 49(2), 285–300.

    Article  Google Scholar 

  11. Grzymała-Busse, J. W., & Rzasa, W. (2008). Local and global approximations for incomplete data. Transactions on Rough Sets, 8, 21–34.

    Google Scholar 

  12. Halpern, J. Y., & Moses, Y. (1990). Knowledge and common knowledge in a distributed environment. Journal of the ACM, 37(3), 549–587.

    Article  Google Scholar 

  13. Hughes, G. E., & Cresswell, M. J. (1996). A new introduction to modal logic. London: Routledge.

    Book  Google Scholar 

  14. Katzberg, J., & Ziarko, W. (1994). Variable precision rough sets with asymmetric bounds. In W. Ziarko (Ed.), Proc. RSKD’93, Canada (pp. 167–177). London: Springer-Verlag.

    Google Scholar 

  15. Khan, M. A., & Banerjee, M. (2008). Formal reasoning with rough sets in multiple-source approximation systems. International Journal of Approximate Reasoning, 49(2), 466–477.

    Article  Google Scholar 

  16. Khan, M. A., & Banerjee, M. (2008). Multiple-source approximation systems: Membership functions and indiscernibility. In G. Wang, et al. (Eds.), Proc. RSKT’08, China. LNAI (Vol. 5009, pp. 80–87). Springer-Verlag.

  17. Liau, C. J. (2000). An overview of rough set semantics for modal and quantifier logics. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 8(1), 93–118.

    Google Scholar 

  18. Margaris, A. (1966). First order mathematical logic. London: Blaisdell Publishing Company.

    Google Scholar 

  19. Murakami, Y. (2005). Modal logic of partitions. Ph.D. thesis, University Graduate School, Indiana University.

  20. Orłowska, E. (1990). Kripke semantics for knowledge representation logics. Studia Logica, 49, 255–272.

    Article  Google Scholar 

  21. Orłowska, E., & Pawlak, Z. (1984). Expressive power of knowledge representation systems. International Journal of Man-Machine Studies, 20(5), 485–500.

    Article  Google Scholar 

  22. Pagliani, P. (2004). Pretopologies and dynamic spaces. Fundamenta Informaticae, 59(2–3), 221–239.

    Google Scholar 

  23. Pawlak, Z. (1982). Rough sets. International Journal of Computer and Information Science, 11(5), 341–356.

    Article  Google Scholar 

  24. Pawlak, Z. (1991). Rough sets. Theoretical aspects of reasoning about data. Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  25. Rasiowa, H., & Marek, W. (1989). On reaching consensus by groups of intelligent agents. In Z. W. Ras (Ed.), Proc. ISMIS’89, Charlotte, N. C. (pp. 234–243). North-Holland, New York.

    Google Scholar 

  26. Rauszer, C. M. (1994). Rough logic for multiagent systems. In M. Masuch & L. Polos (Eds.), Knowledge representation and reasoning under uncertainty. LNAI (Vol. 808, pp. 161–181). Springer-Verlag.

  27. Skowron, A., & Stepaniuk, J. (1996). Tolerance approximation spaces. Fundamenta Informaticae, 27, 245–253.

    Google Scholar 

  28. Ślȩzak, D., & Ziarko, W. (2005). The investigation of the Bayesian rough set model. International Journal of Approximate Reasoning, 40, 81–91.

    Article  Google Scholar 

  29. Vakarelov, D. (1987). Abstract characterization of some knowledge representation systems and the logic NIL of nondeterministic information. In P. Jorrand & V. Sgurev (Eds.), Artificial intelligence II (pp. 255–260). North–Holland.

  30. Vakarelov, D. (1991). Modal logics for knowledge representation systems. Theoretical Computer Science, 90, 433–456.

    Google Scholar 

  31. Wong, S. K. M. (1998). A rough set model for reasoning about knowledge. In L. Polkowski & A. Skowron (Eds.), Rough sets in knowledge discovery 1: Methodology and applications (pp. 276–285). Physica-Verlag.

  32. Yao, Y. Y. (2008). Probabilistic rough set approximations. International Journal of Approximate Reasoning, 49(2), 255–271.

    Article  Google Scholar 

  33. Yao, Y. Y., Wong, S. K. M., & Lin, T. Y. (1997). A review of rough set models. In T. Y. Lin & C. N. Cercone (Eds.), Rough sets and data mining: Analysis for imprecise data (pp. 47–75). Boston: Kluwer.

    Google Scholar 

  34. Ziarko, W. (2008). Probabilistic approach to rough sets. International Journal of Approximate Reasoning, 49(2), 272–284.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, 208016, India

    Md. Aquil Khan & Mohua Banerjee

Authors
  1. Md. Aquil Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohua Banerjee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohua Banerjee.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Khan, M.A., Banerjee, M. A Logic for Multiple-source Approximation Systems with Distributed Knowledge Base. J Philos Logic 40, 663–692 (2011). https://doi.org/10.1007/s10992-010-9163-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10992-010-9163-1

Keywords

Search

Navigation