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Separating Sets by Modal Formulas

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Algebraic Methodology and Software Technology (AMAST 1999)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1548))

Abstract

Subsequently, we introduce a reasoning formalism which in particular allows to express that certain sets in a system of subsets of a given set are disjoint. The main purpose of considering such a family of subsets is to be able to investigate how knowledge grows as subsets shrink in the course of time. We actually introduce a trimodal logic: we have a system containing operators for knowledge and time, of which the latter corresponds to the effort of measurement and reminds of the nexttime operator of temporal logic; an operator separating sets is added then. Socalled subset tree models appear as the relevant semantical structures.We present an axiomatization of the set of valid formulas encompassing the three operators and their interaction. Afterwards the completeness of the given axiomatization is proved. We also give arguments showing that the logic is decidable.

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References

  1. Chellas, B.F.: Modal Logic: An Introduction. Cambridge University Press, Cambridge (1980)

    MATH  Google Scholar 

  2. Artemov, S., Davoren, J., Nerode, A.:. Topological Semantics for Hybrid Systems. In: Adian, S., Nerode, A. (eds.): Logical Foundations of Computer Science, LFCS’97. Lecture Notes in Computer Science, Vol. 1234. Springer-Verlag, Berlin Heidelberg New York (1997) 1–8

    Google Scholar 

  3. Dabrowski, A., Moss, L.S., Parikh, R.: Topological Reasoning and The Logic of Knowledge. Annals of Pure and Applied Logic 78 (1996) 73–110

    Article  MATH  MathSciNet  Google Scholar 

  4. Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning about Knowledge. MIT Press, Cambridge (Mass.) (1995)

    MATH  Google Scholar 

  5. Georgatos, K.: Knowledge Theoretic Properties of Topological Spaces. In: Masuch, M., Polos, L. (eds.): Knowledge Representation and Uncertainty. Lecture Notes in Computer Science, Vol. 808. Springer-Verlag, Berlin Heidelberg New York (1994) 147–159

    Google Scholar 

  6. Georgatos, K.: Reasoning about Knowledge on Computation Trees. In: MacNish, C., Pearce, D., Pereira, L.M. (eds.): Logics in Artificial Intelligence, JELIA’94. Lecture Notes in Computer Science, Vol. 838. Springer-Verlag, Berlin Heidelberg New York (1994) 300–315

    Chapter  Google Scholar 

  7. Georgatos, K.: private communication

    Google Scholar 

  8. Goldblatt, R.: Logics of Time and Computation. Center for the Study of Language and Information, Stanford (1987)

    Google Scholar 

  9. Heinemann, B.: ‘Topological’ Modal Logic of Subset Frames with Finite Descent. In: Kautz, H., Selman, B. (eds.): Proceedings of the Fourth International Symposium on Artificial Intelligence and Mathematics, AI/MATH-96. Fort Lauderdale (1996) 83–86

    Google Scholar 

  10. Heinemann, B.: On binary computation structures. Mathematical Logic Quaterly 43 (1997) 203–215

    Article  MATH  MathSciNet  Google Scholar 

  11. Heinemann, B.: A topological generalization of propositional linear time temporal logic. In: Prívara, I., Ružička, P. (eds.): Mathematical Foundations of Computer Science (22nd International Symposium, MFCS’97). Lecture Notes in Computer Science, Vol. 1295. Springer-Verlag, Berlin Heidelberg New York (1997) 289–297

    Google Scholar 

  12. Heinemann, B.: Including Separation in the Modal Logic of Subset Spaces. Technical Report, No. 231. FernUniversität, Hagen (1998)

    Google Scholar 

  13. Ladner, R.E.: The Computational Complexity of Provability in Systems of Modal Propositional Logic. SIAM Journal of Computing 6 (1977) 467–480

    Article  MATH  MathSciNet  Google Scholar 

  14. Marx, M., Venema, Y.: Multi-Dimensional Modal Logic. Kluwer, Dordrecht (1996)

    Google Scholar 

  15. Moss, L.S., Parikh R.: Topological Reasoning and The Logic of Knowledge. In: Moses, Y. (ed.): Theoretical Aspects of Reasoning about Knowledge, TARK 1992. Morgan Kaufmann, San Francisco (1992) 95–105

    Google Scholar 

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Authors and Affiliations

  1. Fachbereich Informatik, FernUniversität Hagen, D-58084, Hagen, Germany

    Bernhard Heinemann

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  1. Bernhard Heinemann
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Editors and Affiliations

  1. Departamento de Informática, Pontifícia Laboratório de Métodos Formais, Universidade Católica do Rio de Janeiro, Rua Marquês De São Vicente, 225, 22453-900, Rio de Janeiro, RJ, Brazil

    Armando M. Haeberer

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© 1998 Springer-Verlag Berlin Heidelberg

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Heinemann, B. (1998). Separating Sets by Modal Formulas. In: Haeberer, A.M. (eds) Algebraic Methodology and Software Technology. AMAST 1999. Lecture Notes in Computer Science, vol 1548. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-49253-4_12

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  • DOI: https://doi.org/10.1007/3-540-49253-4_12

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-65462-9

  • Online ISBN: 978-3-540-49253-5

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