Abstract
The study of graph games serves as a way to analyze an existing logic, as well as an inspiration for designing new logics. Given the fact that game-theoretic analysis is reviving in AI study, a new stress in the study of graph games should be the performance of standard algorithmic tasks that are conducted on graphs. In this paper, we carry out a case study on the respective graph game for three main local variants of sabotage modal logic, which have a broad range of applications in various other fields. We analyze the solution complexity for each game and show the implications these results have on their corresponding logic. This work is a first attempt to understand why similar-looking variants of a graph game and their corresponding logics can have drastically different computational complexities, with the goal to bring up a more general topic that requires further studies, namely to identify the parameters of games and logic that crucially affect complexity.
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Notes
- 1.
In Areces et al. (2009), it is been proven that the model checking for \(ML(\textcircled {k}, \textcircled {r}, \textcircled {e})\) is PSPACE-complete and since \(ML(\textcircled {k},\textcircled {r})\) (which is, in turn, the same language as \(ML_{\emptyset }\) in Blando et al. (2018)) is a fragment of \(ML(\textcircled {k}, \textcircled {r}, \textcircled {e})\) and thus the upper bound for the model checking for \(ML_{\emptyset }\) is PSPACE, and by the inclusion of expressive power of the three modal languages for poison game and the clearly polynomial-time translation between the languages, all three of the languages has PSPACE as the upper bound for model checking.
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Acknowledgements
This research is supported by Tsinghua University Initiative Scientific Research Program (2017THZWYX08). I wish to thank Johan van Benthem for his invaluable advice throughout the development of this project. I also wish to thank my parents and several teachers and friends, whose company has given me the courage to discover and create, and has made this journey meaningful.
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Zhang, T. (2020). Solution Complexity of Local Variants of Sabotage Game. In: Liu, F., Ono, H., Yu, J. (eds) Knowledge, Proof and Dynamics. Logic in Asia: Studia Logica Library. Springer, Singapore. https://doi.org/10.1007/978-981-15-2221-5_6
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