Skip to main content
SpringerLink
Log in

The Completeness Problem for Modal Logic

  • Conference paper
  • First Online:
Logical Foundations of Computer Science (LFCS 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10703))

Included in the following conference series:

  • 586 Accesses

Abstract

We introduce the completeness problem for Modal Logic and examine its complexity. For a definition of completeness for formulas, given a formula of a modal logic, the completeness problem asks whether the formula is complete for that logic. We discover that completeness and validity have the same complexity — with certain exceptions for which there are, in general, no complete formulas. To prove upper bounds, we present a non-deterministic polynomial-time procedure with an oracle from PSPACE that combines tableaux and a test for bisimulation, and determines whether a formula is complete.

This research was partly supported by the project “TheoFoMon: Theoretical Foundations for Monitorability” (grant number: 163406-051) of the Icelandic Research Fund.

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

Access this chapter

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 EPUB and 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

Institutional subscriptions

Notes

  1. 1.

    According to our definition, for a finite model \(\mathcal {M}=(W,R,V)\) and \(a \in W\), V(a) can be infinite. However, we are mainly interested in \((W,R,V_P)\) for finite sets of propositions P, which justifies calling \(\mathcal {M}\) finite.

  2. 2.

    Although for the purposes of this paper we only consider a specific set of modal logics, it is interesting to note that the corollary can be extended to a much larger class of logics.

  3. 3.

    This is also a corollary of Lemma 4, as these are extensions of D and T.

  4. 4.

    We note that US is different from UP; for UP, if T has an accepting path for x, then it is guaranteed that it has a unique accepting path for x.

References

  1. Ladner, R.E.: The computational complexity of provability in systems of modal propositional logic. SIAM J. Comput. 6(3), 467–480 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  2. Halpern, J.Y., Rêgo, L.C.: Characterizing the NP-PSPACE gap in the satisfiability problem for modal logic. J. Logic Comput. 17(4), 795–806 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Halpern, J.Y., Moses, Y.: A guide to completeness and complexity for modal logics of knowledge and belief. Artif. Intell. 54(3), 319–379 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Artemov, S.: Syntactic epistemic logic. In: Book of Abstracts, 15th Congress of Logic, Methodology and Philosophy of Science CLMPS 2015, pp. 109–110 (2015)

    Google Scholar 

  5. Artemov, S.: Syntactic epistemic logic and games (2016)

    Google Scholar 

  6. Hennessy, M., Milner, R.: Algebraic laws for nondeterminism and concurrency. J. ACM (JACM) 32(1), 137–161 (1985)

    Article  MATH  Google Scholar 

  7. Milner, R.: Communication and Concurrency. Prentice-Hall Inc., Upper Saddle River (1989)

    MATH  Google Scholar 

  8. Graf, S., Sifakis, J.: A modal characterization of observational congruence on finite terms of CCS. Inf. Control 68(1–3), 125–145 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  9. Steffen, B., Ingólfsdóttir, A.: Characteristic formulas for processes with divergence. Inf. Comput. 110(1), 149–163 (1994)

    Article  MATH  Google Scholar 

  10. Mller-Olm, M.: Derivation of characteristic formulae. Electr. Notes Theor. Comput. Sci. 18, 159–170 (1998)

    Article  MathSciNet  Google Scholar 

  11. Aceto, L., Della Monica, D., Fábregas, I., Ingólfsdóttir, A.: When are prime formulae characteristic? In: Italiano, G.F., Pighizzini, G., Sannella, D.T. (eds.) MFCS 2015. LNCS, vol. 9234, pp. 76–88. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48057-1_6

    Chapter  Google Scholar 

  12. Fine, K.: Normal forms in modal logic. Notre Dame J. Formal Logic 16(2), 229–237 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  13. Achilleos, A.: The completeness problem for modal logic. CoRR abs/1605.01004 (2016)

    Google Scholar 

  14. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  15. Chagrov, A., Zakharyaschev, M.: Modal Logic. Oxford University Press, Oxford (1997)

    MATH  Google Scholar 

  16. Paige, R., Tarjan, R.E.: Three partition refinement algorithms. SIAM J. Comput. 16(6), 973–989 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fitting, M.: Tableau methods of proof for modal logics. Notre Dame J. Formal Logic 13(2), 237–247 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  18. Massacci, F.: Single step tableaux for modal logics. J. Autom. Reasoning 24(3), 319–364 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. D’Agostino, M., Gabbay, D.M., Hähnle, R., Posegga, J.: Handbook of Tableau Methods. Springer, Dordrecht (1999). https://doi.org/10.1007/978-94-017-1754-0

    Book  MATH  Google Scholar 

  20. Blass, A., Gurevich, Y.: On the unique satisfiability problem. Inf. Control 55(1–3), 80–88 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  21. Moss, L.S.: Finite models constructed from canonical formulas. J. Philos. Logic 36(6), 605–640 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author is grateful to Luca Aceto for valuable comments that helped improve the quality of this paper.

Author information

Authors and Affiliations

  1. School of Computer Science, Reykjavik University, Reykjavik, Iceland

    Antonis Achilleos

Authors
  1. Antonis Achilleos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Antonis Achilleos .

Editor information

Editors and Affiliations

  1. City University of New York, New York, New York, USA

    Sergei Artemov

  2. Cornell University, Ithaca, New York, USA

    Anil Nerode

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Achilleos, A. (2018). The Completeness Problem for Modal Logic. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2018. Lecture Notes in Computer Science(), vol 10703. Springer, Cham. https://doi.org/10.1007/978-3-319-72056-2_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-72056-2_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-72055-5

  • Online ISBN: 978-3-319-72056-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation