Skip to main content
SpringerLink
Log in
Book cover

International Symposium on Mathematical Foundations of Computer Science

MFCS 2011: Mathematical Foundations of Computer Science 2011 pp 364–375Cite as

A Universally Defined Undecidable Unimodal Logic

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6907))

Abstract

Modal logics are widely used in computer science. The complexity of their satisfiability problems has been an active field of research since the 1970s. We prove that even very “simple” modal logics can be undecidable: We show that there is an undecidable unimodal logic that can be obtained by restricting the allowed models with an equality-free first-order formula in which only universal quantifiers appear.

Supported in part by NSF grant IIS-0713061, the DAAD postdoc program, and by a Friedrich Wilhelm Bessel Research Award. Work done in part while Henning Schnoor was at the Rochester Institute of Technology.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aucher, G., Boella, G., van der Torre, L.: Privacy policies with modal logic: The dynamic turn. In: Governatori, G., Sartor, G. (eds.) DEON 2010. LNCS, vol. 6181, pp. 196–213. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  2. Andréka, H., van Benthem, J., Németi, I.: Modal languages and bounded fragments of predicate logic. Journal of Philosophical Logic 27, 217–274 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Blackburn, P., de Rijke, M., Venema, Y.: Modal logic. Cambridge University Press, New York (2001)

    MATH  Google Scholar 

  4. Bennett, B., Galton, A.: A unifying semantics for time and events. Artificial Intelligence 153(1-2), 13–48 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Blackburn, P., Spaan, E.: A modal perspective on the computational complexity of attribute value grammar. Journal of Logic, Language, and Information 2(2), 129–169 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Demri, S., Konikowska, B.: Relative similarity logics are decidable: Reduction to FO2 with equality. In: Dix, J., Fariñas del Cerro, L., Furbach, U. (eds.) JELIA 1998. LNCS (LNAI), vol. 1489, pp. 279–293. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  7. Frendrup, U., Hüttel, H., Jensen, J.: Modal logics for cryptographic processes. In: Proceedings of EXPRESS 2002 (2002)

    Google Scholar 

  8. Fine, K.: Logics containing K4. Part II. J. Symb. Log. 50(3), 619–651 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  9. Garvon, G., Goranko, V.: Modal logic with names. Journal of Philosophical Logic 22(6), 607–636 (1993)

    Article  MathSciNet  Google Scholar 

  10. Gabelaia, D., Kurucz, A., Wolter, F., Zakharyaschev, M.: Products of ’transitive’ modal logics. J. Symb. Log. 70(3), 993–1021 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Harel, D.: Recurring dominoes: Making the highly undecidable highly understandable (preliminary report). In: Karpinski, M. (ed.) FCT 1983. LNCS, vol. 158, pp. 177–194. Springer, Heidelberg (1983)

    Google Scholar 

  12. Hemaspaandra, E.: The price of universality. Notre Dame Journal of Formal Logic 37(2), 174–203 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Halpern, J., Moses, Y., Tuttle, M.: A knowledge-based analysis of zero knowledge. In: STOC 1988: Proceedings of the 20th Annual ACM Symposium on Theory of Computing, pp. 132–147. ACM Press, New York (1988)

    Chapter  Google Scholar 

  14. Hemaspaandra, E., Schnoor, H.: On the complexity of elementary modal logics. In: Albers, S., Weil, P. (eds.) STACS, Germany, LIPIcs, vol. 1, pp. 349–360. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany (2008)

    Google Scholar 

  15. Hemaspaandra, E., Schnoor, H.: A simplest undecidable modal logic. CoRR, abs/1105.0845 (2011)

    Google Scholar 

  16. Hemaspaandra, E., Schnoor, H., Schnoor, I.: Generalized modal satisfiability. J. Comput. Syst. Sci. 76(7), 561–578 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Halpern, J., Vardi, M.: The complexity of reasoning about knowledge and time. I. Lower bounds. J. Comput. Syst. Sci. 38(1), 195–237 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ladner, R.: The computational complexity of provability in systems of modal propositional logic. SIAM Journal of Computation 6(3), 467–480 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ladner, R., Reif, J.: The logic of distributed protocols. In: Halpern, J.Y. (ed.) TARK, pp. 207–222. Morgan Kaufmann, San Francisco (1986)

    Google Scholar 

  20. Moody, J.: Modal logic as a basis for distributed computation. Technical Report CMU-CS-03-194, School of Computer Science, Carnegie Mellon University (October 2003)

    Google Scholar 

  21. Reynolds, M., Zakharyaschev, M.: On the products of linear modal logics. J. Log. Comput. 11(6), 909–931 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. Spaan, E.: Nexttime is not necessary. In: Parikh, R. (ed.) TARK, pp. 241–256. Morgan Kaufmann, San Francisco (1990)

    Google Scholar 

  23. Spaan, E.: Complexity of Modal Logics. PhD thesis, Department of Mathematics and Computer Science, University of Amsterdam (1993)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Rochester Institute of Technology, Rochester, NY, 14623, U.S.A.

    Edith Hemaspaandra

  2. Institute for Computer Science, Christian-Albrechts-Universität zu Kiel, Kiel, Germany

    Henning Schnoor

Authors
  1. Edith Hemaspaandra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Henning Schnoor
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Informatics, University of Warsaw, ul. Banacha 2, 02-097, Warsaw, Poland

    Filip Murlak  & Piotr Sankowski  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag GmbH Berlin Heidelberg

About this paper

Cite this paper

Hemaspaandra, E., Schnoor, H. (2011). A Universally Defined Undecidable Unimodal Logic. In: Murlak, F., Sankowski, P. (eds) Mathematical Foundations of Computer Science 2011. MFCS 2011. Lecture Notes in Computer Science, vol 6907. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22993-0_34

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-22993-0_34

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22992-3

  • Online ISBN: 978-3-642-22993-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation