Skip to main content

On the Complexity of Graded Modal Logics with Converse

  • 597 Accesses

Part of the Lecture Notes in Computer Science book series (LNAI,volume 11468)

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-19570-0_42
  • Chapter length: 17 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   89.00
Price excludes VAT (USA)
  • ISBN: 978-3-030-19570-0
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   119.99
Price excludes VAT (USA)
Fig. 1.
Fig. 2.
Fig. 3.

Notes

  1. 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].

References

  1. Baader, F., Horrocks, I., Lutz, C., Sattler, U.: An Introduction to Description Logic. Cambridge University Press, Cambridge (2017). https://doi.org/10.1017/9781139025355

    CrossRef  MATH  Google Scholar 

  2. Bednarczyk, B., Kieronski, E., Witkowski, P.: On the complexity of graded modal logics with converse. CoRR abs/1812.04413 (2018). http://arxiv.org/abs/1812.04413

  3. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, New York (2001). https://doi.org/10.1017/CBO9781107050884

  4. Chen, C.-C., Lin, I.-P.: The complexity of propositional modal theories and the complexity of consistency of propositional modal theories. In: Nerode, A., Matiyasevich, Y.V. (eds.) LFCS 1994. LNCS, vol. 813, pp. 69–80. Springer, Heidelberg (1994). https://doi.org/10.1007/3-540-58140-5_8

    CrossRef  Google Scholar 

  5. Demri, S., de Nivelle, H.: Deciding regular grammar logics with converse through first-order logic. J. Logic Lang. Inf. 14(3), 289–329 (2005). https://doi.org/10.1007/s10849-005-5788-9

    MathSciNet  CrossRef  MATH  Google Scholar 

  6. Gutiérrez-Basulto, V., Ibáñez-García, Y.A., Jung, J.C.: Number restrictions on transitive roles in description logics with nominals. In: Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, San Francisco, California, USA, 4–9 February 2017, pp. 1121–1127 (2017)

    Google Scholar 

  7. Kazakov, Y., Pratt-Hartmann, I.: A note on the complexity of the satisfiability problem for graded modal logics. In: Proceedings of the 24th Annual IEEE Symposium on Logic in Computer Science, LICS 2009, Los Angeles, CA, USA, 11–14 August 2009, pp. 407–416 (2009). https://doi.org/10.1109/LICS.2009.17

  8. Kazakov, Y., Sattler, U., Zolin, E.: How many legs do I have? Non-simple roles in number restrictions revisited. In: 2007 Proceedings of 14th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning, LPAR 2007, Yerevan, Armenia, 15–19 October, pp. 303–317 (2007). https://doi.org/10.1007/978-3-540-75560-9_23

  9. Ladner, R.E.: The computational complexity of provability in systems of modal propositional logic. SIAM J. Comput. 6(3), 467–480 (1977). https://doi.org/10.1137/0206033

    MathSciNet  CrossRef  MATH  Google Scholar 

  10. Blackburn, P., van Benthem, J.: Handbook of Modal Logic, chapter Modal Logic: A Semantic Perspective, pp. 255–325. Elsevier (2006)

    Google Scholar 

  11. Pratt-Hartmann, I.: Complexity of the two-variable fragment with counting quantifiers. J. Logic Lang. Inf. 14(3), 369–395 (2005). https://doi.org/10.1007/s10849-005-5791-1

    MathSciNet  CrossRef  MATH  Google Scholar 

  12. Pratt-Hartmann, I.: Complexity of the guarded two-variable fragment with counting quantifiers. J. Log. Comput. 17(1), 133–155 (2007). https://doi.org/10.1093/logcom/exl034

    MathSciNet  CrossRef  MATH  Google Scholar 

  13. Pratt-Hartmann, I.: On the computational complexity of the numerically definite syllogistic and related logics. Bull. Symbolic Logic 14(1), 1–28 (2008). https://doi.org/10.2178/bsl/1208358842

    MathSciNet  CrossRef  MATH  Google Scholar 

  14. Tobies, S.: PSPACE reasoning for graded modal logics. J. Log. Comput. 11(1), 85–106 (2001). https://doi.org/10.1093/logcom/11.1.85

    MathSciNet  CrossRef  MATH  Google Scholar 

  15. Zolin, E.: Undecidability of the transitive graded modal logic with converse. J. Log. Comput. 27(5), 1399–1420 (2017). https://doi.org/10.1093/logcom/exw026

    MathSciNet  CrossRef  MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to Bartosz Bednarczyk , Emanuel Kieroński or Piotr Witkowski .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

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

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-19570-0_42

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-19569-4

  • Online ISBN: 978-3-030-19570-0

  • eBook Packages: Computer ScienceComputer Science (R0)