Merging Frameworks for Interaction

A variety of logical frameworks have been developed to study rational agents interacting over time. This paper takes a closer look at one particular interface, between two systems that both address the dynamics of knowledge and information flow. The first is Epistemic Temporal Logic (ETL) which uses linear or branching time models with added epistemic structure induced by agents’ different capabilities for observing events. The second framework is Dynamic Epistemic Logic (DEL) that describes interactive processes in terms of epistemic event models which may occur inside modalities of the language. This paper systematically and rigorously relates the DEL framework with the ETL framework. The precise relationship between DEL and ETL is explored via a new representation theorem characterizing the largest class of ETL models corresponding to DEL protocols in terms of notions of Perfect Recall, No Miracles, and Bisimulation Invariance. We then focus on new issues of completeness. One contribution is an axiomatization for the dynamic logic of public announcements constrained by protocols, which has been an open problem for some years, as it does not fit the usual ‘reduction axiom’ format of DEL. Finally, we provide a number of examples that show how DEL suggests an interesting fine-structure inside ETL.

[1]  J. Benthem.  One is a Lonely Number: on the logic of communication , 2003 .

[2]  Emiliano Lorini,et al.  The cognitive structure of surprise: looking for basic principles , 2007 .

[3]  Alex Pentland,et al.  Facing the future , 2003, 2003 IEEE International SOI Conference. Proceedings (Cat. No.03CH37443).

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

[5]  M. de Rijke,et al.  Logic, language and computation, vol. 2 , 1999 .

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

[7]  Wiebe van der Hoek,et al.  Dynamic Epistemic Logic and Knowledge Puzzles , 2007, ICCS.

[8]  Johan van Benthem,et al.  Merging frameworks for interaction: DEL and ETL , 2007, TARK '07.

[9]  Alasdair Urquhart,et al.  Temporal Logic , 1971 .

[10]  Johan van Benthem,et al.  Epistemic dynamics and protocol information , 2009 .

[11]  Lawrence S. Moss,et al.  The Undecidability of Iterated Modal Relativization , 2005, Stud Logica.

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

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

[14]  Johan van Benthem,et al.  The Tree of Knowledge in Action: Towards a Common Perspective , 2006, Advances in Modal Logic.

[15]  D. Gabbay,et al.  Many-Dimensional Modal Logics: Theory and Applications , 2003 .

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

[17]  Frank Wolter,et al.  Handbook of Modal Logic , 2007, Studies in logic and practical reasoning.

[18]  Richard Spencer-Smith,et al.  Modal Logic , 2007 .

[19]  J. Gerbrandy.  Bisimulations on Planet Kripke , 1999 .

[20]  Joseph Y. Halpern,et al.  The complexity of reasoning about knowledge and time , 1986, STOC '86.

[21]  Andreas Herzig,et al.  What can we achieve by arbitrary announcements?: A dynamic take on Fitch's knowability , 2007, TARK '07.

[22]  Carsten Lutz,et al.  Complexity and succinctness of public announcement logic , 2006, AAMAS '06.

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

[24]  Joseph Y. Halpern,et al.  The Complexity of Reasoning about Knowledge and Time. I. Lower Bounds , 1989, J. Comput. Syst. Sci..

[25]  Joseph Y. Halpern,et al.  Complete Axiomatizations for Reasoning about Knowledge and Time , 2002, SIAM J. Comput..

[26]  Radha Jagadeesan,et al.  Games and Full Completeness for Multiplicative Linear Logic , 1994, J. Symb. Log..

[27]  J. Benthem,et al.  Diversity of Logical Agents in Games , 2004 .

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

[29]  Maricarmen Martinez,et al.  The Stories of Logic and Information , 2006 .

[30]  Ramaswamy Ramanujam,et al.  A Knowledge Based Semantics of Messages , 2003, J. Log. Lang. Inf..

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

[32]  Eric Pacuit,et al.  Some comments on history based structures , 2007, J. Appl. Log..

[33]  Kevin T. Kelly.  The Logic of Reliable Inquiry , 1996 .

[34]  J. Benthem,et al.  Logics of communication and change , 2006 .

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

[36]  Johan van Benthem,et al.  Modal Frame Correspondences and Fixed-Points , 2006, Stud Logica.

[37]  Giacomo Bonanno,et al.  Memory and perfect recall in extensive games , 2004, Games Econ. Behav..

[38]  Peter Øhrstrøm,et al.  Temporal Logic , 1994, Lecture Notes in Computer Science.

[39]  Rohit Parikh,et al.  Reasoning about Communication Graphs , 2007 .

[40]  Joshua Sack,et al.  Temporal Languages for Epistemic Programs , 2008, J. Log. Lang. Inf..