Convergence, Continuity and Recurrence in Dynamic Epistemic Logic

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10455)


The paper analyzes dynamic epistemic logic from a topological perspective. The main contribution consists of a framework in which dynamic epistemic logic satisfies the requirements for being a topological dynamical system thus interfacing discrete dynamic logics with continuous mappings of dynamical systems. The setting is based on a notion of logical convergence, demonstratively equivalent with convergence in Stone topology. Presented is a flexible, parametrized family of metrics inducing the latter, used as an analytical aid. We show maps induced by action model transformations continuous with respect to the Stone topology and present results on the recurrent behavior of said maps.


Dynamic epistemic logic Limit behavior Convergence Recurrence Dynamical systems Metric spaces General topology Modal logic 



The contribution of R.K. Rendsvig was funded by the Swedish Research Council through the Knowledge in a Digital World project and by The Center for Information and Bubble Studies, sponsored by The Carlsberg Foundation. The contribution of D. Klein was partially supported by the Deutsche Forschungsgemeinschaft (DFG) and Grantová agentura České republiky (GAČR) as part of the joint project From Shared Evidence to Group Attitudes [RO 4548/6-1].


  1. 1.
    Aucher, G.: Generalizing AGM to a multi-agent setting. Logic J. IGPL 18(4), 530–558 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baltag, A., Moss, L.S.: Logics for epistemic programs. Synthese 139(2), 165–224 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Baltag, A., Moss, L.S., Solecki, S.: The logic of public announcements, common knowledge, and private suspicions. In: TARK 1998. Morgan Kaufmann (1998)Google Scholar
  4. 4.
    Baltag, A., Renne, B.: Dynamic epistemic logic. In: The Stanford Encyclopedia of Philosophy (2016). Fall 2016th EditionGoogle Scholar
  5. 5.
    Baltag, A., Smets, S.: A qualitative theory of dynamic interactive belief revision. In: Proceedings of LOFT 7. Amsterdam University Press (2008)Google Scholar
  6. 6.
    Baltag, A., Smets, S.: Group belief dynamics under iterated revision: fixed points and cycles of joint upgrades. In: TARK 2009. ACM (2009)Google Scholar
  7. 7.
    van Benthem, J.: Games in dynamic-epistemic logic. Bull. Econ. Res. 53(1), 219–249 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    van Benthem, J.: “One is a Lonely Number”: logic and communication. In: Logic Colloquium 2002. Lecture Notes in Logic, vol. 27. Association for Symbolic Logic (2006)Google Scholar
  9. 9.
    van Benthem, J.: Dynamic logic for belief revision. J. Appl. Non-Class. Logics 17(2), 129–155 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    van Benthem, J.: Logical Dynamics of Information and Interaction. Cambridge University Press, Cambridge (2011)CrossRefzbMATHGoogle Scholar
  11. 11.
    van Benthem, J.: Oscillations, logic, and dynamical systems. In: Ghosh, S., Szymanik, J. (eds.) The Facts Matter. College Publications, London (2016)Google Scholar
  12. 12.
    van Benthem, J., van Eijck, J., Kooi, B.: Logics of communication and change. Inf. Comput. 204(11), 1620–1662 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    van Benthem, J., Gerbrandy, J., Hoshi, T., Pacuit, E.: Merging frameworks for interaction. J. Philos. Logic 38(5), 491–526 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2001)CrossRefzbMATHGoogle Scholar
  15. 15.
    Bolander, T., Birkegaard, M.: Epistemic planning for single- and multi-agent systems. J. Appl. Non-Class. Logics 21(1), 9–34 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bolander, T., Jensen, M., Schwarzentruber, F.: Complexity results in epistemic planning. In: Proceedings of IJCAI 2015. AAAI Press (2015)Google Scholar
  17. 17.
    Caridroit, T., Konieczny, S., de Lima, T., Marquis, P.: On distances between KD45n Kripke models and their use for belief revision. In: ECAI 2016. IOS Press (2016)Google Scholar
  18. 18.
    Dégremont, C.: The temporal mind: observations on the logic of belief change in interactive systems. Ph.D. thesis, University of Amsterdam (2010)Google Scholar
  19. 19.
    van Ditmarsch, H., Kooi, B.: Semantic results for ontic and epistemic change. In: Logic and the Foundations of Game and Decision Theory (LOFT 7). Texts in Logic and Games, vol. 3. Amsterdam University Press (2008)Google Scholar
  20. 20.
    van Ditmarsch, H., van der Hoek, W., Kooi, B.: Dynamic Epistemic Logic. Springer, Dordrecht (2008). doi: 10.1007/978-1-4020-5839-4 zbMATHGoogle Scholar
  21. 21.
    Eisner, T., Farkas, B., Haase, M., Nagel, R.: Operator Theoretic Aspects of Ergodic Theory. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-16898-2 CrossRefzbMATHGoogle Scholar
  22. 22.
    Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning About Knowledge. The MIT Press, Cambridge (1995)zbMATHGoogle Scholar
  23. 23.
    Fernández-Duque, D.: A sound and complete axiomatization for dynamic topological logic. J. Symb. Logic 77(3), 1–26 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Fernández-Duque, D.: Dynamic topological logic of metric spaces. J. Symb. Logic 77(1), 308–328 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Girard, P., Seligman, J., Liu, F.: General dynamic dynamic logic. In: Bolander, T., Brauner, T., Ghilardi, S., Moss, L. (eds.) Advances in modal logics, vol. 9. College Publications, London (2012)Google Scholar
  26. 26.
    Goranko, V.: Logical topologies and semantic completeness. In: Logic Colloquium 1999. Lecture Notes in Logic, vol. 17. AK Peters (2004)Google Scholar
  27. 27.
    Goranko, V., Otto, M.: Model theory of modal logic. In: Blackburn, P., van Benthem, J., Wolter, F. (eds.) Handbook of Modal Logic. Elsevier, Amsterdam (2007)Google Scholar
  28. 28.
    Halpern, Y.J., Moses, Y.: Knowledge and common knowledge in a distributed environment. J. ACM 37(3), 549–587 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Hasselblatt, B., Katok, A.: Principal structures. In: Hasselblatt, B., Katok, A. (eds.) Handbook of Dynamical Systems, vol. 1A. Elsevier, Amsterdam (2002)Google Scholar
  30. 30.
    Hintikka, J.: Knowledge and Belief: An Introduction to the Logic of the Two Notions. College Publications, London (1962). 2nd, 2005 EditionGoogle Scholar
  31. 31.
    Klein, D., Rendsvig, R.K.: Metrics for Formal Structures, with an Application to Kripke Models and their Dynamics. arXiv:1704.00977 (2017)
  32. 32.
    Klein, D., Rendsvig, R.K.: Turing Completeness of Finite, Epistemic Programs. arXiv:1706.06845 (2017)
  33. 33.
    Kooi, B., Renne, B.: Generalized arrow update logic. In: TARK 2011. ACM, New York (2011)Google Scholar
  34. 34.
    Kremer, P., Mints, G.: Dynamical topological logic. Bull. Symb. Logic 3, 371–372 (1997)Google Scholar
  35. 35.
    Kremer, P., Mints, G.: Dynamic Topological Logic. In: Aiello, M., Pratt-Hartmann, I., Van Benthem, J. (eds.) Handbook of Spatial Logics. Springer, Dordrecht (2007). doi: 10.1007/978-1-4020-5587-4_10 Google Scholar
  36. 36.
    Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  37. 37.
    Munkres, J.R.: Topology, 2nd edn. Prentice-Hall, Englewood Cliffs (2000)zbMATHGoogle Scholar
  38. 38.
    Plaza, J.A.: Logics of public communications. In: Proceedings of the 4th International Symposium on Methodologies for Intelligent Systems (1989)Google Scholar
  39. 39.
    Rendsvig, R.K.: Towards a theory of semantic competence. Master’s thesis, Department of Philosophy and Science Studies and Department of Mathematics, Roskilde University (2011)Google Scholar
  40. 40.
    Rendsvig, R.K.: Diffusion, influence and best-response dynamics in networks: an action model approach. In: Proceedings of ESSLLI 2014 Student Session arXiv:1708.01477 (2014)
  41. 41.
    Rendsvig, R.K.: Pluralistic ignorance in the bystander effect: Informational dynamics of unresponsive witnesses in situations calling for intervention. Synthese 191(11), 2471–2498 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Rendsvig, R.K.: Model transformers for dynamical systems of dynamic epistemic logic. In: Hoek, W., Holliday, W.H., Wang, W. (eds.) LORI 2015. LNCS, vol. 9394, pp. 316–327. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-48561-3_26 CrossRefGoogle Scholar
  43. 43.
    Sadzik, T.: Exploring the iterated update universe. ILLC PP-2006-26 (2006)Google Scholar
  44. 44.
    de Vries, J.: Topological Dynamical Systems. de Gruyter, Berlin (2014)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of PhilosophyBayreuth UniversityBayreuthGermany
  2. 2.Department of Political ScienceUniversity of BambergBambergGermany
  3. 3.Theoretical PhilosophyLund UniversityLundSweden
  4. 4.Center for Information and Bubble StudiesUniversity of CopenhagenCopenhagenDenmark

Personalised recommendations