Abstract
A complete classification of the complexity of the local and global satisfiability problems for graded modal language over traditional classes of frames has already been established. By “traditional” classes of frames we mean those characterized by any positive combination of reflexivity, seriality, symmetry, transitivity, and the Euclidean property. In this paper we fill the gaps remaining in an analogous classification of the graded modal language with graded converse modalities. In particular, we show its NExpTime-completeness over the class of Euclidean frames, demonstrating this way that over this class the considered language is harder than the language without graded modalities or without converse modalities. We also consider its variation disallowing graded converse modalities, but still admitting basic converse modalities. Our most important result for this variation is confirming an earlier conjecture that it is decidable over transitive frames. This contrasts with the undecidability of the language with graded converse modalities.
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Notes
- 1.
As explained to the first author by Emil Jeřábek, the latter bound can be alternatively proved by a reduction from TB, whose ExpTime-hardness follows from [4].
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Acknowledgements
We thank Evgeny Zolin for providing us a comprehensive list of gaps in the classification of the complexity of graded modal logics and for sharing with us his tikz files with modal cubes. We also thank Emil Jeřábek for his explanations concerning
. B.B. is supported by the Polish Ministry of Science and Higher Education program “Diamentowy Grant” no. DI2017 006447. E.K. and P.W. are supported by Polish National Science Centre grant no. 2016/21/B/ST6/01444.
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Bednarczyk, B., Kieroński, E., Witkowski, P. (2019). On the Complexity of Graded Modal Logics with Converse. In: Calimeri, F., Leone, N., Manna, M. (eds) Logics in Artificial Intelligence. JELIA 2019. Lecture Notes in Computer Science(), vol 11468. Springer, Cham. https://doi.org/10.1007/978-3-030-19570-0_42
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