The Complexity of Non-Iterated Probabilistic Justification Logic

  • Ioannis KokkinisEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9616)


The logic \(\mathsf {PJ}\) is a probabilistic logic defined by adding (non-iterated) probability operators to the basic justification logic \(\mathsf {J}\). In this paper we establish upper and lower bounds for the complexity of the derivability problem in the logic \(\mathsf {PJ}\). The main result of the paper is that the complexity of the derivability problem in \(\mathsf {PJ}\) remains the same as the complexity of the derivability problem in the underlying logic \(\mathsf {J}\), which is \(\varPi _2^p\)-complete. This implies hat the probability operators do not increase the complexity of the logic, although they arguably enrich the expressiveness of the language.


Justification logic Probabilistic logic Complexity Derivability Satisfiability 



The author is grateful to Antonis Achilleos, Thomas Studer and the anonymous referees for valuable comments and remarks that helped him improve the quality of the paper substantially.


  1. 1.
    Achilleos, A.: Nexp-completeness and universal hardness results for justification logic, cSR 2015, pp. 27–52 (2015)Google Scholar
  2. 2.
    Artemov, S.N.: Operational modal logic. Technical report, MSI 95–29, Cornell University, December 1995Google Scholar
  3. 3.
    Artemov, S.N.: Explicit provability and constructive semantics. Bull. Symbolic Logic 7(1), 1–36 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Artemov, S.N.: The ontology of justifications in the logical setting. Studia Logica 100(1–2), 17–30 (2012). Published online, February 2012MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Artemov, S.N., Fitting, M.: Justification logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. Fall 2012 edn. (2012).
  6. 6.
    Bucheli, S., Kuznets, R., Studer, T.: Justifications for common knowledge. J. Appl. Non-Classical Logics 21(1), 35–60 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bucheli, S., Kuznets, R., Studer, T.: Partial realization in dynamic justification logic. In: Beklemishev, L.D., de Queiroz, R. (eds.) WoLLIC 2011. LNCS, vol. 6642, pp. 35–51. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  8. 8.
    Buss, S.R., Kuznets, R.: Lower complexity bounds in justification logic. Ann. Pure Appl. Logic 163(7), 888–905 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chvátal, V.: Linear programming. W. H. Freeman and Company, New York (1983)zbMATHGoogle Scholar
  10. 10.
    Fagin, R., Halpern, J., Megiddo, N.: A logic for reasoning about probabilities. Inf. Comput. 87, 78–128 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fan, T., Liau, C.: A logic for reasoning about justified uncertain beliefs. In: Yang, Q., Wooldridge, M. (eds.) Proceedings of IJCAI 2015, pp. 2948–2954. AAAI Press (2015)Google Scholar
  12. 12.
    Ghari, M.: Justification logics in a fuzzy setting. ArXiv e-prints, July 2014Google Scholar
  13. 13.
    Keisler, J.: Hyperfinite model theory. In: Gandy, R.O., Hyland, J.M.E. (eds.) Logic Colloquim 1976, p. 510. North-Holland (1977)Google Scholar
  14. 14.
    Kokkinis, I.: On the complexity of probabilistic justification logic. ArXiv e-prints (2015)Google Scholar
  15. 15.
    Kokkinis, I., Maksimović, P., Ognjanović, Z., Studer, T.: First steps towards probabilistic justification logic. Logic J. IGPL 23(4), 662–687 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kokkinis, I., Ognjanović, Z., Studer, T.: Probabilistic justification logic. In: Artemov, S., Nerode, A. (eds.) Symposium on Logical Foundations in Computer Science 2016 (2016, to appear)Google Scholar
  17. 17.
    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)CrossRefGoogle Scholar
  18. 18.
    Kuznets, R.: Complexity Issues in Justification Logic. Ph.D. thesis, City University of New York, May 2008.
  19. 19.
    Kuznets, R., Studer, T.: Justifications, ontology, and conservativity. In: Bolander, T., Braüner, T., Ghilardi, S., Moss, L. (eds.) Advances in Modal Logic, vol. 9, pp. 437–458. College Publications, London (2012)Google Scholar
  20. 20.
    Milnikel, R.S.: Derivability in certain subsystems of the logic of proofs is \(\Pi ^p_2\)-complete. Ann. Pure Appl. Logic 145(3), 223–239 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Milnikel, R.S.: The logic of uncertain justifications. Ann. Pure Appl. Logic 165(1), 305–315 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Nilsson, N.: Probabilistic logic. Artif. Intell. 28, 7187 (1986)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ognjanović, Z., Rašković, M., Marković, Z.: Probability logics. Logic Comput. Sci. 12(20), 35–111 (2009). Zbornik radova subseriesMathSciNetzbMATHGoogle Scholar
  24. 24.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)zbMATHGoogle Scholar

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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institute of Computer ScienceUniversity of BernBernSwitzerland

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