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Convergence, Continuity and Recurrence in Dynamic Epistemic Logic

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Logic, Rationality, and Interaction (LORI 2017)

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Abstract

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.

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Notes

  1. 1.

    \(\varvec{\mathcal {L}_{\varLambda }}\) is isomorphic to the domain of the Lindenbaum algebra of \(\varLambda \).

  2. 2.

    That all models in X are image-finite is a sufficient condition, cf. the Hennessy-Milner Theorem. See e.g. [14] or [27].

  3. 3.

    Space does not allow for a discussion of the remaining metrics of [1, 17], but see [31].

  4. 4.

    A logic \(\varLambda \) is logically compact if any arbitrary set A of formulas is \(\varLambda \)-consistent iff every finite subset of A is \(\varLambda \)-consistent.

  5. 5.

    An \(\varvec{\mathcal {L}}_{\varLambda }\) modal space \(\varvec{X}\) is saturated iff for each \(\varLambda \)-consistent set of formulas A, there is an \(\varvec{x}\in \varvec{X}\) such that \(x\vDash A\). Saturation relates to the notion of strong completeness, cf. e.g. [14, Proposition 4.12]. See [31] for its use in a more general context.

  6. 6.

    Multi-pointed action models are also referred to as epistemic programs in [2], and allow encodings akin to knowledge-based programs [22] of interpreted systems, cf. [42].

  7. 7.

    The precondition of \(\sigma \) specify the conditions under which \(\sigma \) is executable, while its postcondition may dictate the posterior values of a finite, possibly empty, set of atoms.

  8. 8.

    I.e. a conjuction of literals, where a literal is an atom or a negated atom.

  9. 9.

    Or \(\omega \)-limit point. The \(\omega \) is everywhere omitted as time here only moves forward.

  10. 10.

    We paraphrase van Benthem and Sadzik using the terminology introduced.

  11. 11.

    See [16] for an elegant and generalizing exposition.

  12. 12.

    The details differ depending on whether \(\varSigma {\scriptstyle \varGamma }\) must be static, but non-Boolean for Proposition 10, or Boolean, but non-static for Proposition 11. See resp. [15, 32].

  13. 13.

    For Proposition 11, tape cell content may be encoded using atomic propositions, changeable through postconditions, cf. [15]; for Proposition 10, cell content is written by adding and removing additional states, cf. [32].

  14. 14.

    The exact form is straightforward from the constructions used in [15, 32].

  15. 15.

    A similar argument shows that all \(x^{Z}\) with \(Z\subseteq \mathbb {N}\) co-infinite are recurrent points. Hence \(\omega _{f}(\varvec{x}'_{k})\) for any \(\varvec{x}'_{k}\in (\varvec{x}'_{n})_{n\in \mathbb {N}_{0}}\) contains uncountably many recurrent points.

  16. 16.

    Hence also the multi-agent belief revision policies lexicographic upgrade and elite change, also known as radical and conservative upgrade, introduced in [9], cf. [5].

  17. 17.

    In the omitted part of the quotation from the introduction.

References

  1. Aucher, G.: Generalizing AGM to a multi-agent setting. Logic J. IGPL 18(4), 530–558 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baltag, A., Moss, L.S.: Logics for epistemic programs. Synthese 139(2), 165–224 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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. Baltag, A., Renne, B.: Dynamic epistemic logic. In: The Stanford Encyclopedia of Philosophy (2016). Fall 2016th Edition

    Google Scholar 

  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. 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. van Benthem, J.: Games in dynamic-epistemic logic. Bull. Econ. Res. 53(1), 219–249 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  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. van Benthem, J.: Dynamic logic for belief revision. J. Appl. Non-Class. Logics 17(2), 129–155 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. van Benthem, J.: Logical Dynamics of Information and Interaction. Cambridge University Press, Cambridge (2011)

    Book  MATH  Google Scholar 

  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. van Benthem, J., van Eijck, J., Kooi, B.: Logics of communication and change. Inf. Comput. 204(11), 1620–1662 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. van Benthem, J., Gerbrandy, J., Hoshi, T., Pacuit, E.: Merging frameworks for interaction. J. Philos. Logic 38(5), 491–526 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  15. Bolander, T., Birkegaard, M.: Epistemic planning for single- and multi-agent systems. J. Appl. Non-Class. Logics 21(1), 9–34 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bolander, T., Jensen, M., Schwarzentruber, F.: Complexity results in epistemic planning. In: Proceedings of IJCAI 2015. AAAI Press (2015)

    Google Scholar 

  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. 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. 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. van Ditmarsch, H., van der Hoek, W., Kooi, B.: Dynamic Epistemic Logic. Springer, Dordrecht (2008). doi:10.1007/978-1-4020-5839-4

    MATH  Google Scholar 

  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

    Book  MATH  Google Scholar 

  22. Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning About Knowledge. The MIT Press, Cambridge (1995)

    MATH  Google Scholar 

  23. Fernández-Duque, D.: A sound and complete axiomatization for dynamic topological logic. J. Symb. Logic 77(3), 1–26 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Fernández-Duque, D.: Dynamic topological logic of metric spaces. J. Symb. Logic 77(1), 308–328 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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. Goranko, V.: Logical topologies and semantic completeness. In: Logic Colloquium 1999. Lecture Notes in Logic, vol. 17. AK Peters (2004)

    Google Scholar 

  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. Halpern, Y.J., Moses, Y.: Knowledge and common knowledge in a distributed environment. J. ACM 37(3), 549–587 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  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. Hintikka, J.: Knowledge and Belief: An Introduction to the Logic of the Two Notions. College Publications, London (1962). 2nd, 2005 Edition

    Google Scholar 

  31. Klein, D., Rendsvig, R.K.: Metrics for Formal Structures, with an Application to Kripke Models and their Dynamics. arXiv:1704.00977 (2017)

  32. Klein, D., Rendsvig, R.K.: Turing Completeness of Finite, Epistemic Programs. arXiv:1706.06845 (2017)

  33. Kooi, B., Renne, B.: Generalized arrow update logic. In: TARK 2011. ACM, New York (2011)

    Google Scholar 

  34. Kremer, P., Mints, G.: Dynamical topological logic. Bull. Symb. Logic 3, 371–372 (1997)

    Google Scholar 

  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. Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  37. Munkres, J.R.: Topology, 2nd edn. Prentice-Hall, Englewood Cliffs (2000)

    MATH  Google Scholar 

  38. Plaza, J.A.: Logics of public communications. In: Proceedings of the 4th International Symposium on Methodologies for Intelligent Systems (1989)

    Google Scholar 

  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. 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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Chapter  Google Scholar 

  43. Sadzik, T.: Exploring the iterated update universe. ILLC PP-2006-26 (2006)

    Google Scholar 

  44. de Vries, J.: Topological Dynamical Systems. de Gruyter, Berlin (2014)

    Book  MATH  Google Scholar 

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Acknowledgements

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].

Author information

Authors and Affiliations

  1. Department of Philosophy, Bayreuth University, Bayreuth, Germany

    Dominik Klein

  2. Department of Political Science, University of Bamberg, Bamberg, Germany

    Dominik Klein

  3. Theoretical Philosophy, Lund University, Lund, Sweden

    Rasmus K. Rendsvig

  4. Center for Information and Bubble Studies, University of Copenhagen, Copenhagen, Denmark

    Rasmus K. Rendsvig

Authors
  1. Dominik Klein
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  2. Rasmus K. Rendsvig
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Correspondence to Dominik Klein or Rasmus K. Rendsvig .

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Editors and Affiliations

  1. Institute for Logic, Language and Computation, University of Amsterdam , Amsterdam, Noord-Holland, The Netherlands

    Alexandru Baltag

  2. Department of Philosophy, The University of Auckland, Auckland, Auckland, New Zealand

    Jeremy Seligman

  3. Graduate School of Letters, Hokkaido University , Sapporo, Japan

    Tomoyuki Yamada

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Klein, D., Rendsvig, R.K. (2017). Convergence, Continuity and Recurrence in Dynamic Epistemic Logic. In: Baltag, A., Seligman, J., Yamada, T. (eds) Logic, Rationality, and Interaction. LORI 2017. Lecture Notes in Computer Science(), vol 10455. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55665-8_8

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