Studia Logica

, Volume 104, Issue 4, pp 597–640 | Cite as

The Undecidability of Quantified Announcements



This paper demonstrates the undecidability of a number of logics with quantification over public announcements: arbitrary public announcement logic (APAL), group announcement logic (GAL), and coalition announcement logic (CAL). In APAL we consider the informative consequences of any announcement, in GAL we consider the informative consequences of a group of agents (this group may be a proper subset of the set of all agents) all of which are simultaneously (and publicly) making known announcements. So this is more restrictive than APAL. Finally, CAL is as GAL except that we now quantify over anything the agents not in that group may announce simultaneously as well. The logic CAL therefore has some features of game logic and of ATL. We show that when there are multiple agents in the language, the satisfiability problem is undecidable for APAL, GAL, and CAL. In the single agent case, the satisfiability problem is decidable for all three logics.


Dynamic Epistemic Logic Complexity of modal logics 


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  1. 1.
    Ågotnes T.: Action and knowledge in alternating-time temporal logic. Synthese (Special Section on Knowledge, Rationality and Action) 149(2), 377–409 (2006)Google Scholar
  2. 2.
    Ågotnes T., Balbiani P., van Ditmarsch H., Seban P.: Group announcement logic. Journal of Applied Logic 8, 62–81 (2010)CrossRefGoogle Scholar
  3. 3.
    Ågotnes, T., and H. van Ditmarsch, Coalitions and announcements, in Proceedings of the of 7th AAMAS, IFAAMAS, Columbia, 2008, pp. 673–680.Google Scholar
  4. 4.
    Ågotnes, T., H. van Ditmarsch, and T. French, The undecidability of group announcements, in Proceedings of the of the 13th AAMAS, 2014, pp. 893–900.Google Scholar
  5. 5.
    Alur R., Henzinger T. A., Kupferman O.: Alternating-time temporal logic. Journal of the ACM 49, 672–713 (2002)CrossRefGoogle Scholar
  6. 6.
    Balbiani P., Baltag A., van Ditmarsch H., Herzig A., Hoshi T., Lima T. D.: ‘Knowable’ as ‘known after an announcement’. Review of Symbolic Logic 1(3), 305–334 (2008)CrossRefGoogle Scholar
  7. 7.
    Baltag, A., L. Moss, and S. Solecki, The logic of public announcements, common knowledge, and private suspicions, in Proceedings of the 7th TARK, Morgan Kaufmann, Burlington, 1998, pp. 43–56.Google Scholar
  8. 8.
    Belardinelli, F., and W. van der Hoek, Epistemic quantified boolean logic: Expressiveness and completeness results, in Q. Yang and M. Wooldridge (eds.), Proceedings of 24th IJCAI, AAAI Press, New York, 2015, pp. 2748–2754.Google Scholar
  9. 9.
    Berger, R., The Undecidability of the Domino Problem, Number 66 in Memoirs of the American Mathematical Society, American Mathematical Society, 1966.Google Scholar
  10. 10.
    Bolander T., Andersen M.: Epistemic planning for single and multi-agent systems. Journal of Applied Non-classical Logics 21(1), 9–34 (2011)CrossRefGoogle Scholar
  11. 11.
    Bolander, T., M. Jensen, and F. Schwarzentruber, Complexity results in epistemic planning, in Q. Yang and M. Wooldridge (eds.), Proceedings of 24th IJCAI, AAAI Press, New York, 2015, pp. 2791–2797.Google Scholar
  12. 12.
    Bozzelli L., van Ditmarsch H., French T., Hales J., Pinchinat S.: Refinement modal logic. Information and Computation 239, 303–339 (2014)CrossRefGoogle Scholar
  13. 13.
    Charrier, T., and F. Schwarzentruber, Arbitrary public announcement logic with mental programs, in G. Weiss, P. Yolum, R. Bordini, and E. Elkind (eds.), Proceedings of AAMAS, ACM, New York, 2015, pp. 1471–1479.Google Scholar
  14. 14.
    Cordón-Franco A., van Ditmarsch H., Fernández-Duque D., Soler-Toscano F.: A colouring protocol for the generalized russian cards problem. Theoretical Computer Science 495, 81–95 (2013)CrossRefGoogle Scholar
  15. 15.
    Fagin, R., J. Halpern, Y. Moses, and M. Vardi, Reasoning About Knowledge, MIT Press, Cambridge, MA, 1995.Google Scholar
  16. 16.
    Fischer M., Ladner R.: Propositional dynamic logic of regular programs. Journal of Computer and System Sciences 18(2), 194–211 (1979)CrossRefGoogle Scholar
  17. 17.
    French, T., and H. van Ditmarsch, Undecidability for arbitrary public announcement logic. in C. Areces and R. Goldblatt (eds.), Advances in Modal Logic 7, College Publications, London, 2008, pp. 23–42.Google Scholar
  18. 18.
    Gerbrandy, J., Bisimulations on Planet Kripke. PhD thesis, University of Amsterdam, 1999. ILLC Dissertation Series DS-1999-01.Google Scholar
  19. 19.
    Hales, J., Arbitrary action model logic and action model synthesis, in Proceedings of the 28th LICS, IEEE, 2013, pp. 253–262.Google Scholar
  20. 20.
    Harel D.: Effective transformations on infinite trees, with applications to high undecidability, dominoes, and fairness. Journal of the ACM 33(1), 224–248 (1986)CrossRefGoogle Scholar
  21. 21.
    Lutz, C., Complexity and succinctness of public announcement logic, in Proceedings of the 5th AAMAS, 2006, pp. 137–144.Google Scholar
  22. 22.
    Miller J., Moss L.: The undecidability of iterated modal relativization. Studia Logica 79(3), 373–407 (2005)CrossRefGoogle Scholar
  23. 23.
    Pauly M.: A modal logic for coalitional power in games. Journal of Logic and Computation 12(1), 149–166 (2002)CrossRefGoogle Scholar
  24. 24.
    Plaza, J., Logics of public communications, in Proceedings of the 4th ISMIS, Oak Ridge National Laboratory, Oak Ridge, 1989, pp. 201–216.Google Scholar
  25. 25.
    van Benthem J.: Correspondence Theory. Springer, Berlin (1984)CrossRefGoogle Scholar
  26. 26.
    van Benthem, J., Logics for information update, in J. van Benthem (ed.), Proceedings of TARK VIII, Morgan Kaufmann, Los Altos, 2001, pp. 51–88.Google Scholar
  27. 27.
    van Benthem, J., One is a lonely number: on the logic of communication. in Logic Colloquium 2002, Lecture Notes in Logic, vol. 27. A. K. Peters, Wellesley, 2006, pp. 96–129.Google Scholar
  28. 28.
    van Benthem, J., Logic in Games, MIT Press, Cambridge, MA, 2014.Google Scholar
  29. 29.
    van der Hoek, W., and M. Wooldridge, Tractable multiagent planning for epistemic goals, in Proceedings of the First International Joint Conference on Autonomous Agents & Multiagent Systems (AAMAS 2002), ACM, New York, 2002, pp. 1167–1174.Google Scholar
  30. 30.
    van Ditmarsch, H., Quantifying notes, in Proceedings of 19th WoLLIC, LNCS 7456, Springer, New York, 2012, pp. 89–109.Google Scholar
  31. 31.
    van Ditmarsch, H., T. French, and S. Pinchinat, Future event logic—axioms and complexity. in L. Beklemishev, V. Goranko, and V. Shehtman (eds.), Advances in Modal Logic 8, College Publications, London, 2010, pp. 77–99.Google Scholar
  32. 32.
    van Ditmarsch, H., W. van der Hoek, and B. Kooi, Dynamic Epistemic Logic, vol. 337 of Synthese Library, Springer, New York, 2007.Google Scholar

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© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.University of BergenBergenNorway
  2. 2.LORIA CNRS - Université de LorraineNancyFrance
  3. 3.The University of Western AustraliaPerthAustralia

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