Abstract
We introduce a modal logic in which one of the operators expresses properties of points (knowledge states) in a neighbourhood of a given point (the actual set of alternatives), and other operators speak about shrinking such neighbourhoods gradually (representing the change of the set of states in time). Based on a modification of the topological language due to Moss and Parikh [Moss and Parikh 1992] we generalize a certain fragment of prepositional branching time logic to the logic of knowledge in this way. To keep the notation simple we confine ourselves to binary branching. Thus we define a trimodal logic comprising one knowledge-operator and two nexttime-operators. The formulas are interpreted in binary ramified subset tree models. We present an axiomatization of the set T of theorems valid for this class of semantical domains and prove its completeness as the main result of the paper. Furthermore, decidability of T is shown, and its complexity is determined.
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References
Chellas, B. F. 1980. Modal Logic: An Introduction. Cambridge: Cambridge University Press.
Dabrowski, A., L. S. Moss, and R. Parikh. 1996. Topological Reasoning and The Logic of Knowledge. Ann. Pure Appl. Logic 78:73–110.
Fagin, R., J. Y. Halpern, Y. Moses, and M. Y. Vardi. 1995. Reasoning about Knowledge. Cambridge(Mass.): MIT Press.
Gabbay, D. M. 1976. Investigations in Modal and Tense Logics with Applications to Problems in Philosophy and Linguistics. Dordrecht: Reidel
Georgatos, K. 1994. Reasoning about Knowledge on Computation Trees. In Proc. Logics in Artificial Intelligence (JELIA '94), eds. C. MacNish, D. Pearce, and L. M. Pereira, 300–315. Springer. LNCS 838.
Goldblatt, R. 1987. Logics of Time and Computation. CSLI Lecture Notes Number 7. Stanford: Center for the Study of Language and Information.
Heinemann, B. 1996a. 'Topological’ Aspects of Knowledge and Nexttime. Informatik Berichte 209. Hagen: Fernuniversität, December.
Heinemann, B. 1996b. ‘Topological’ Modal Logic of Subset Frames with Finite Descent. In Proc. 4th Intern. Symp. on Artificial and Mathematics, AI/MATH-96, 83–86. Fort Lauderdale.
Heinemann, B. 1997a. On the Complexity of Prefix Formulas in Modal Logic of Subset Spaces. In Logical Foundations of Computer Science, LFCS'97, eds. S. Adian and A. Nerode. Springer, to appear.
Heinemann, B. 1997b. Topological Nexttime Logic. In Advances in Modal Logic '96, eds. M. Kracht, M. de Rijke, H. Wansing, and M. Zakharyaschev. Kluwer. to appear.
Ladner, R. E. 1977. The Computational Complexity of Provability in Systems of Modal Prepositional Logic. SIAM J. Comput. 6:467–480.
Moss, L. S., and R. Parikh. 1992. Topological Reasoning and The Logic of Knowledge. In Proc. 4th Conf. on Theoretical Aspects of Reasoning about Knowledge (TARK 1992), ed. Y. Moses, 95–105. Morgan Kaufmann.
Weihrauch, K. 1995. A Foundation of Computable Analysis. EATCS Bulletin 57.
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Heinemann, B. (1997). A modal logic for reasoning about knowledge and time on binary subset trees. In: Gabbay, D.M., Kruse, R., Nonnengart, A., Ohlbach, H.J. (eds) Qualitative and Quantitative Practical Reasoning. FAPR ECSQARU 1997 1997. Lecture Notes in Computer Science, vol 1244. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0035630
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DOI: https://doi.org/10.1007/BFb0035630
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