Convergence, continuity, recurrence and Turing completeness in dynamic epistemic logic1

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 Stone topology, 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. Among the recurrence results, we show maps induced by finite action models may have uncountably many recurrent points, even when initiated on a finite input model. Several recurrence results draws on the class of action models being Turing complete, for which the paper provides proof in the postcondition-free case. As upper bounds, is shown that either 1 atom, 3 agents and preconditions of modal depth 18, or 1 atom, 7 agents and preconditions of modal depth 3 suffices for Turing completeness.

[1]  J. van Benthem,et al.  Oscillations, Logic, and Dynamical Systems , 2015 .

[2]  M. Stone.  The theory of representations for Boolean algebras , 1936 .

[3]  M. de Rijke,et al.  Modal Logic , 2001, Cambridge Tracts in Theoretical Computer Science.

[4]  David Fernández-Duque.  Dynamic topological logic of metric spaces , 2012, J. Symb. Log..

[5]  Itay Kaplan,et al.  Forking and Dividing in NTP2 theories , 2012, The Journal of Symbolic Logic.

[6]  Johan van Benthem,et al.  "ONE IS A LONELY NUMBER": LOGIC AND COMMUNICATION , 2006 .

[7]  Rasmus K. Rendsvig,et al.  Pluralistic ignorance in the bystander effect: informational dynamics of unresponsive witnesses in situations calling for intervention , 2014, Synthese.

[8]  Barteld P. Kooi,et al.  Generalized arrow update logic , 2011, TARK XIII.

[9]  Thomas Bolander,et al.  Epistemic planning for single- and multi-agent systems , 2011, J. Appl. Non Class. Logics.

[10]  Thomas Caridroit,et al.  On Distances Between KD45n Kripke Models and Their Use for Belief Revision , 2016, ECAI.

[11]  Ronald Fagin,et al.  Reasoning about knowledge , 1995 .

[12]  Jaakko Hintikka,et al.  Knowledge and Belief: An Introduction to the Logic of the Two Notions. , 1965 .

[13]  François Schwarzentruber,et al.  Complexity Results in Epistemic Planning , 2015, IJCAI.

[14]  Lawrence S. Moss,et al.  The Logic of Public Announcements and Common Knowledge and Private Suspicions , 1998, TARK.

[15]  Hans van Ditmarsch,et al.  Semantic results for ontic and epistemic change , 2006, ArXiv.

[16]  Stefan Friedrich,et al.  Topology , 2019, Arch. Formal Proofs.

[17]  Dominik Klein,et al.  Converging on Common Knowledge , 2019, IJCAI.

[18]  Rasmus K. Rendsvig,et al.  Model Transformers for Dynamical Systems of Dynamic Epistemic Logic , 2015, LORI.

[19]  Rasmus K. Rendsvig,et al.  Towards a Theory of Semantic Competence , 2011 .

[20]  Guillaume Aucher,et al.  Generalizing AGM to a multi-agent setting , 2010, Log. J. IGPL.

[21]  T. Eisner,et al.  Operator Theoretic Aspects of Ergodic Theory , 2015 .

[22]  Alexandru Baltag,et al.  Group belief dynamics under iterated revision: fixed points and cycles of joint upgrades , 2009, TARK '09.

[23]  Alexandru Baltag,et al.  A qualitative theory of dynamic interactive belief revision , 2008 .

[24]  Lawrence S. Moss,et al.  Logics for Epistemic Programs , 2004, Synthese.

[25]  Johan van Benthem,et al.  Logical Dynamics of Information and Interaction , 2014 .

[26]  W. Hoek,et al.  Dynamic Epistemic Logic , 2007 .

[27]  Jan van Eijck,et al.  Logical topologies and semantic completeness , 2016 .

[28]  Rasmus K. Rendsvig,et al.  Logical Dynamics and Dynamical Systems , 2018 .

[29]  Jan A. Plaza,et al.  Logics of public communications , 2007, Synthese.

[30]  Rasmus K. Rendsvig,et al.  Diffusion, Influence and Best-Response Dynamics in Networks: An Action Model Approach , 2017, ArXiv.

[31]  Philip Kremer,et al.  Dynamic topological logic , 2005, Ann. Pure Appl. Log..

[32]  Yurii Rogozhin,et al.  Small Universal Turing Machines , 1996, Theor. Comput. Sci..

[33]  Valentin Goranko,et al.  Model theory of modal logic , 2007, Handbook of Modal Logic.

[34]  J. Benthem.  Games in dynamic epistemic logic , 2001 .

[35]  Jan van Eijck,et al.  Logics of communication and change , 2006, Inf. Comput..

[36]  M. Stone,et al.  The Theory of Representation for Boolean Algebras , 1936 .

[37]  C. Degrémont.  The temporal mind : observations on the logic of belief change in interactive systems , 2010 .

[38]  J. Bell.  STONE SPACES (Cambridge Studies in Advanced Mathematics 3) , 1987 .

[39]  Joseph Y. Halpern,et al.  Knowledge and common knowledge in a distributed environment , 1984, JACM.

[40]  Rasmus K. Rendsvig,et al.  Convergence, Continuity and Recurrence in Dynamic Epistemic Logic , 2017, LORI.

[41]  Jelle Gerbrandy,et al.  Dynamic epistemic logic , 1998 .

[42]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[43]  Fenrong Liu,et al.  General Dynamic Dynamic Logic , 2012, Advances in Modal Logic.

[44]  Johan van Benthem,et al.  Merging Frameworks for Interaction , 2009, J. Philos. Log..

[45]  Tomasz Sadzik,et al.  Exploring the Iterated Update Universe , 2005 .

[46]  Johan van Benthem,et al.  Dynamic logic for belief revision , 2007, J. Appl. Non Class. Logics.

[47]  T. Eisner,et al.  Topological Dynamical Systems , 2022, Dimension Groups and Dynamical Systems.

[48]  Douglas Lind,et al.  An Introduction to Symbolic Dynamics and Coding , 1995 .