Archive for Mathematical Logic

, Volume 53, Issue 7–8, pp 949–967 | Cite as

On the complexity of the closed fragment of Japaridze’s provability logic

  • Fedor Pakhomov


We consider the well-known provability logic GLP. We prove that the GLP-provability problem for polymodal formulas without variables is PSPACE-complete. For a number n, let \({L^{n}_0}\) denote the class of all polymodal variable-free formulas without modalities \({\langle n \rangle,\langle n+1\rangle,...}\). We show that, for every number n, the GLP-provability problem for formulas from \({L^{n}_0}\) is in PTIME.


Provability logic Computational complexity Closed fragment 

Mathematics Subject Classification

03F45 03D15 


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  1. 1.
    Beklemishev L.D.: Reflection principles and provability algebras in formal arithmetic. Russ. Math. Surv. 60, 197–268 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Beklemishev, L.D.: Veblen hierarchy in the context of provability algebras. In: Logic, Methodology and Philosophy of Science, Proceedings of the Twelfth International Congress, pp. 65–78. Kings College Publications (2005)Google Scholar
  3. 3.
    Beklemishev L.D., Fernández-Duque D., Joosten J.J.: On provability logics with linearly ordered modalities. Studia Logica 102(3), 541–566 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Beklemishev L.D., Joosten J.J., Vervoort M.: A finitary treatment of the closed fragment of Japaridze’s provability logic. J. Logic Comput. 15(4), 447–463 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bou F., Joosten J.J.: The closed fragment of il is pspace hard. Electron. Notes Theor. Comput. Sci. 278, 47–54 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chagrov, A.V., Rybakov, M.N.: How many variables does one need to prove PSPACE-hardness of modal logics. In: Advances in Modal Logic, pp. 71–82 (2002)Google Scholar
  7. 7.
    Cook S.A., Reckhow R.A.: Time bounded random access machines. J. Comput. Syst. Sci. 7(4), 354–375 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dashkov E.: On the positive fragment of the polymodal provability logic GLP. Math. Notes 91, 318–333 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ignatiev K.N.: On strong provability predicates and the associated modal logics. J. Symb. Log. 58(1), 249–290 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Japaridze, G.K.: The modal logical means of investigation of provability. Thesis in Philosophy, Moscow (in Russian) (1986)Google Scholar
  11. 11.
    Joosten, J.J.: Interpretability formalized. Ph.D thesis, Utrecht University (2004)Google Scholar
  12. 12.
    Ladner R.E.: The computational complexity of provability in systems of modal propositional logic. SIAM J. Comput. 6(3), 467–480 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Shapirovsky, I.: PSPACE-decidability of Japaridze’s polymodal logic. In :Advances in Modal Logic, pp. 289–304 (2008)Google Scholar
  14. 14.
    Stockmeyer, L.J., Meyer, A.R.: Word problems requiring exponential time (preliminary report). In: Proceedings of the Fifth Annual ACM Symposium on Theory of Computing, STOC ’73, pp. 1–9, New York, NY, USA, ACM (1973)Google Scholar
  15. 15.
    Švejdar V.: The decision problem of provability logic with only one atom. Arch. Math. Logic 42(8), 763–768 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteMoscowRussia

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