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.

[1]  Johan van Benthem,et al.  Logic in Games , 2014 .

[2]  James Hales,et al.  Arbitrary Action Model Logic and Action Model Synthesis , 2013, 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science.

[3]  Robert L. Berger.  The undecidability of the domino problem , 1966 .

[4]  Fernando Soler-Toscano,et al.  A colouring protocol for the generalized Russian cards problem , 2012, Theor. Comput. Sci..

[5]  Johan van Benthem.  Logics for information update , 2001 .

[6]  Philippe Balbiani,et al.  Group announcement logic , 2010, J. Appl. Log..

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

[8]  Tim French,et al.  The undecidability of group announcements , 2014, AAMAS.

[9]  Hans van Ditmarsch,et al.  Coalitions and announcements , 2008, AAMAS.

[10]  Sophie Pinchinat,et al.  Future Event Logic - Axioms and Complexity , 2010, Advances in Modal Logic.

[11]  Barteld Kooi,et al.  Dynamic Epistemic Logic , 2013 .

[12]  David Harel,et al.  Effective transformations on infinite trees, with applications to high undecidability, dominoes, and fairness , 1986, JACM.

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

[14]  Andreas Herzig,et al.  ‘KNOWABLE’ AS ‘KNOWN AFTER AN ANNOUNCEMENT’ , 2008, The Review of Symbolic Logic.

[15]  François Schwarzentruber,et al.  Arbitrary Public Announcement Logic with Mental Programs , 2015, AAMAS.

[16]  Richard E. Ladner,et al.  Propositional Dynamic Logic of Regular Programs , 1979, J. Comput. Syst. Sci..

[17]  Hans van Ditmarsch,et al.  Quantifying Notes , 2012, WoLLIC.

[18]  Sophie Pinchinat,et al.  Refinement modal logic , 2012, Inf. Comput..

[19]  Thomas Ågotnes,et al.  Action and Knowledge in Alternating-Time Temporal Logic , 2006, Synthese.

[20]  Tim French,et al.  Undecidability for arbitrary public announcement logic , 2008, Advances in Modal Logic.

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

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

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

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

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

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

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

[28]  Thomas A. Henzinger,et al.  Alternating-time temporal logic , 1999 .

[29]  Marc Pauly,et al.  A Modal Logic for Coalitional Power in Games , 2002, J. Log. Comput..

[30]  Wiebe van der Hoek,et al.  Epistemic Quantified Boolean Logic: Expressiveness and Completeness Results , 2015, IJCAI.

[31]  Michael Wooldridge,et al.  Tractable multiagent planning for epistemic goals , 2002, AAMAS '02.