FAPR 1997, ECSQARU 1997: Qualitative and Quantitative Practical Reasoning pp 298-310

# A modal logic for reasoning about knowledge and time on binary subset trees

• Bernhard Heinemann
Accepted Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1244)

## 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.

## Keywords

Modal Logic Canonical Model Logical Language Semantical Domain Cantor Space
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## References

1. [Chellas 1980]
Chellas, B. F. 1980. Modal Logic: An Introduction. Cambridge: Cambridge University Press.Google Scholar
2. [Dabrowski et al. 1996]
Dabrowski, A., L. S. Moss, and R. Parikh. 1996. Topological Reasoning and The Logic of Knowledge. Ann. Pure Appl. Logic 78:73–110.Google Scholar
3. [Fagin et al. 1995]
Fagin, R., J. Y. Halpern, Y. Moses, and M. Y. Vardi. 1995. Reasoning about Knowledge. Cambridge(Mass.): MIT Press.Google Scholar
4. [Gabbay 1976]
Gabbay, D. M. 1976. Investigations in Modal and Tense Logics with Applications to Problems in Philosophy and Linguistics. Dordrecht: ReidelGoogle Scholar
5. [Georgatos 1994]
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.Google Scholar
6. [Goldblatt 1987]
Goldblatt, R. 1987. Logics of Time and Computation. CSLI Lecture Notes Number 7. Stanford: Center for the Study of Language and Information.Google Scholar
7. [Heinemann 1996a]
Heinemann, B. 1996a. 'Topological’ Aspects of Knowledge and Nexttime. Informatik Berichte 209. Hagen: Fernuniversität, December.Google Scholar
8. [Heinemann 1996b]
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.Google Scholar
9. [Heinemann 1997a]
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.Google Scholar
10. [Heinemann 1997b]
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.Google Scholar
Ladner, R. E. 1977. The Computational Complexity of Provability in Systems of Modal Prepositional Logic. SIAM J. Comput. 6:467–480.Google Scholar
12. [Moss and Parikh 1992]
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.Google Scholar
13. [Weihrauch 1995]
Weihrauch, K. 1995. A Foundation of Computable Analysis. EATCS Bulletin 57.Google Scholar