The NP-Completeness of Reflected Fragments of Justification Logics

  • Samuel R. Buss
  • Roman Kuznets
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5407)

Abstract

Justification Logic studies epistemic and provability phenomena by introducing justifications/proofs into the language in the form of justification terms. Pure justification logics serve as counterparts of traditional modal epistemic logics, and hybrid logics combine epistemic modalities with justification terms. The computational complexity of pure justification logics is typically lower than that of the corresponding modal logics. Moreover, the so-called reflected fragments, which still contain complete information about the respective justification logics, are known to be in NP for a wide range of justification logics, pure and hybrid alike. This paper shows that, under reasonable additional restrictions, these reflected fragments are NP-complete, thereby proving a matching lower bound.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AK06]
    Artemov, S.N., Kuznets, R.: Logical omniscience via proof complexity. In: Ésik, Z. (ed.) CSL 2006. LNCS, vol. 4207, pp. 135–149. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  2. [Art95]
    Artemov, S.N.: Operational modal logic. Technical Report MSI 95–29, Cornell University (December 1995)Google Scholar
  3. [Art01]
    Artemov, S.N.: Explicit provability and constructive semantics. Bulletin of Symbolic Logic 7(1), 1–36 (2001)MathSciNetCrossRefMATHGoogle Scholar
  4. [Art06]
    Artemov, S.N.: Justified common knowledge. Theoretical Computer Science 357(1-3), 4–22 (2006)MathSciNetCrossRefMATHGoogle Scholar
  5. [Art08a]
    Artemov, S.N.: The logic of justification. Technical Report TR–2008010, CUNY Ph.D. Program in Computer Science (September 2008)Google Scholar
  6. [Art08b]
    Artemov, S.N.: Why do we need Justification Logic? Technical Report TR–2008014, CUNY Ph.D. Program in Computer Science (September 2008)Google Scholar
  7. [BB93]
    Bonet, M.L., Buss, S.R.: The deduction rule and linear and near-linear proof simulations. Journal of Symbolic Logic 58(2), 688–709 (1993)MathSciNetCrossRefMATHGoogle Scholar
  8. [Bre00]
    Brezhnev, V.N.: On explicit counterparts of modal logics. Technical Report CFIS 2000–05, Cornell University (2000)Google Scholar
  9. [Kru03]
    Krupski, N.V.: On the complexity of the reflected logic of proofs. Technical Report TR–2003007, CUNY Ph.D. Program in Computer Science (May 2003)Google Scholar
  10. [Kru06]
    Krupski, N.V.: On the complexity of the reflected logic of proofs. Theoretical Computer Science 357(1-3), 136–142 (2006)MathSciNetCrossRefMATHGoogle Scholar
  11. [Kuz00]
    Kuznets, R.: On the complexity of explicit modal logics. In: Clote, P.G., Schwichtenberg, H. (eds.) CSL 2000. LNCS, vol. 1862, pp. 371–383. Springer, Heidelberg (2000); Errata concerning the explicit counterparts of \(\mathcal{D}\) and \(\mathcal{D}4\) are published as [Kuz08b]CrossRefGoogle Scholar
  12. [Kuz08a]
    Kuznets, R.: Complexity Issues in Justification Logic. PhD thesis, CUNY Graduate Center (May 2008)Google Scholar
  13. [Kuz08b]
    Kuznets, R.: Complexity through tableaux in justification logic. In: Abstracts of Plenary Talks, Tutorials, Special Sessions, Contributed Talks of Logic Colloquium (LC 2008), Bern, Switzerland, pp. 38–39 (Abstract) (July 3-8, 2008)Google Scholar
  14. [Kuz08c]
    Kuznets, R.: Self-referentiality of justified knowledge. In: Hirsch, E.A., Razborov, A.A., Semenov, A., Slissenko, A. (eds.) CSR 2008. LNCS, vol. 5010, pp. 228–239. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  15. [Lad77]
    Ladner, R.E.: The computational complexity of provability in systems of modal propositional logic. SIAM Journal on Computing 6(3), 467–480 (1977)MathSciNetCrossRefMATHGoogle Scholar
  16. [Mil07]
    Milnikel, R.: Derivability in certain subsystems of the Logic of Proofs is \(\Pi^p_2\)-complete. Annals of Pure and Applied Logic 145(3), 223–239 (2007)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Samuel R. Buss
    • 1
  • Roman Kuznets
    • 2
  1. 1.Department of MathematicsUniversity of CaliforniaSan DiegoUSA
  2. 2.Institut für Informatik und angewandte MathematikUniversität BernBernSwitzerland

Personalised recommendations