Abstract
Dynamic Epistemic Logic is a logic that is aimed at formally expressing how a person’s knowledge changes. We provide a cut-free labelled sequent calculus (\(\mathbf {GDEL}\)) on the background of existing studies of Hilbert-style axiomatization \(\mathbf {HDEL}\) by Baltag et al. (1989) and labelled calculi for Public Announcement Logic by Maffezioli et al. (2011) and Nomura et al. (2015). We first show that the cut rule is admissible in \(\mathbf {GDEL}\). Then we show \(\mathbf {GDEL}\) is sound and complete for Kripke semantics. Lastly, we touch briefly on our on-going work of an automated theorem prover of \(\mathbf {GDEL}\).
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Notes
- 1.
Labelled sequent calculus is one of the most prevailing methods of sequent calculus for modal logic (cf. Negri et al. [12]).
- 2.
- 3.
Ditmarsch et al. [6] includes union of events \(\mathsf {a^{M}}\cup \mathsf {a'^{M'}}\) in the language, but we do not include it since \([\mathsf {a^{M}}\cup \mathsf {a'^{M'}}]A\) can be handled as a defined connective by \([\mathsf {a^{M}}]A \wedge [\mathsf {a'^{M'}}]A\).
- 4.
This counter-example of the soundness theorem with s-valid is pointed out already in [15, Proposition4], and the proposition of PAL is also applicable to DEL.
- 5.
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Acknowledgement
We would like to thank anonymous reviewers for their constructive comments to our manuscript. The first author would like to thank his superviser, Prof. Satoshi Tojo, for his helpful comments to a draft. Finally, We thank Sean Arn for his proofreading of the final version of the paper. The first author is supported by Grant-in-Aid for JSPS Fellows in pursuing the present research, the third author was supported by JSPS KAKENHI, Grant-in-Aid for Young Scientists 15K21025. The first and third authors were also supported by JSPS Core-to-Core Program (A. Advanced Research Networks).
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Nomura, S., Ono, H., Sano, K. (2016). A Cut-Free Labelled Sequent Calculus for Dynamic Epistemic Logic. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2016. Lecture Notes in Computer Science(), vol 9537. Springer, Cham. https://doi.org/10.1007/978-3-319-27683-0_20
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