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Completing the Picture: Complexity of Graded Modal Logics with Converse

Published online by Cambridge University Press:  27 April 2021

Computational Logic Group, TU Dresden, Dresden, Germany and Institute of Computer Science, University of Wrocław, Wrocław, Poland (e-mail:
Institute of Computer Science, University of Wrocław, Wrocław, Poland (e-mails:,
Institute of Computer Science, University of Wrocław, Wrocław, Poland (e-mails:,
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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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© The Author(s), 2021. Published by Cambridge University Press



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 thank Emil Jeřábek for his explanations concerning K5(, ). We also thank Tomasz Gogacz and Filip Murlak for comments concerning Section 4. Finally, we thank the anonymous reviewers for their useful comments and remarks. Bartosz Bednarczyk is supported by Polish Ministry of Science and Higher Education program “Diamentowy Grant” no. DI2017 006447. Emanuel Kieroński and Piotr Witkowski are supported by Polish National Science Centre grant no. 2016/21/B/ST6/01444.


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