Studia Logica

, Volume 71, Issue 1, pp 1–30 | Cite as

Completeness of Certain Bimodal Logics for Subset Spaces

  • M. Angela Weiss
  • Rohit Parikh

Abstract

Subset Spaces were introduced by L. Moss and R. Parikh in [8]. These spaces model the reasoning about knowledge of changing states.

In [2] a kind of subset space called intersection space was considered and the question about the existence of a set of axioms that is complete for the logic of intersection spaces was addressed. In [9] the first author introduced the class of directed spaces and proved that any set of axioms for directed frames also characterizes intersection spaces.

We give here a complete axiomatization for directed spaces. We also show that it is not possible to reduce this set of axioms to a finite set.

Multi-modal logics logic of knowledge topological reasoning 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Bull, R and K. Segerberg, ‘Basic modal logic’, in D. Gabbay and F. Guenthner, editor, Handbook of Philophical Logic, vol II, pages 1-88. Reidel Publishing Co., 1984.Google Scholar
  2. [2]
    Dabrowski, A., A. Moss and R. Parikh, ‘Topological reasoning and the logic of knowledge’, Annals of Pure and Applied Logic78:73-110, 1996.Google Scholar
  3. [3]
    Georgatos, K., Knowledge in a Spatial Setting. PhD thesis, CUNY, 1993.Google Scholar
  4. [4]
    Georgatos, K., ‘Knowledge theoretical properties for topological spaces’, in M. Masuch and L. Polos, editor, Lecture Notes in Artificial Intelligence, vol 808, pages 147-159. Springer, 1994.Google Scholar
  5. [5]
    Georgatos, K., ‘Reasoning about knowledge on computation trees’, in C. MacNish, D. Pearce, and L. M. Pereira, editor, Logics in Artificial Intelligence, JELIA '94, pages 300-315. Springer, 1994.Google Scholar
  6. [6]
    Heinemann, B., ‘Topological modal logic of subset frames with finite descent’, in Proceedings of the Fourth International Symposium on Artificial and Mathematics, AI/MATH-96, pages 83-86, 1996.Google Scholar
  7. [7]
    Heinemann, B., Subset Space Logics and Logics of Knowledge and Time. PhD thesis, Fachbereich Informatic, Fern Universitat Hagen, 1997.Google Scholar
  8. [8]
    Moss, L. S., and R. Parikh, ‘Topological reasoning and the logic of knowledge’, in Y. Moses, editor, Theoretical Aspects of Reasoning about Knowledge, TARK, pages 95-105. Morgan Kaufmann, 1992.Google Scholar
  9. [9]
    Weiss, M. A., ‘Two classes of subset spaces logically equivalent to intersection spaces’, in submission.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • M. Angela Weiss
    • 1
  • Rohit Parikh
    • 2
  1. 1.Instituto de Matemática e Estatística USPSão PauloBrazil
  2. 2.Brooklyn College and CUNY — Graduate CenterUSA

Personalised recommendations