Only Knowing Meets Common Knowledge

Only knowing captures the intuitive notion that the beliefs of an agent are precisely those that follow from its knowledge base. While only knowing has a simple possible-world semantics in a single agent setting, the many agent case has turned out to be much more challenging. In a recent paper, we proposed an account which arguably extends only knowing to multiple agents in a natural way. However, the approach was limited in that the semantics cannot deal with infinitary notions such as common knowledge. In this work, we lift that serious limitation to obtain a first-order language with only knowing and common knowledge, allowing us to study the interaction between these notions for the very first time. By adding a simple form of public announcement, we then demonstrate how the muddy children puzzle can be cast in terms of logical implications given what is only known initially.

[1]  Joseph Y. Halpern,et al.  Towards a Theory of Knowledge and Ignorance: Preliminary Report , 1989, NMR.

[2]  Sarit Kraus,et al.  Knowledge, Belief and Time , 1986, ICALP.

[3]  Arild Waaler,et al.  Consistency Proofs for Systems of Multi?agent Only Knowing , 2004, Advances in Modal Logic.

[4]  Saul A. Kripke,et al.  Semantical Considerations on Modal Logic , 2012 .

[5]  Wiebe van der Hoek,et al.  A General Approach to Multi-Agent Minimal Knowledge: With Tools and Samples , 2002, Stud Logica.

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

[7]  G. E. Hughes,et al.  An introduction to modal logic, 2e éd., 1 vol , 1973 .

[8]  Gerhard Lakemeyer,et al.  Levesque's Axiomatization of only Knowing is Incomplete , 1995, Artif. Intell..

[9]  Danny Dolev,et al.  Cheating husbands and other stories: A case study of knowledge, action, and communication , 1986, Distributed Computing.

[10]  Michael Wooldridge,et al.  Logics for Multiagent Systems , 2012, AI Mag..

[11]  Jennifer J. Elgot-Drapkin,et al.  Step-Logic and the Three-Wise-Men Problem , 1991, AAAI.

[12]  Ian Pratt-Hartmann,et al.  Total Knowledge , 2000, AAAI/IAAI.

[13]  Vaishak Belle,et al.  Multi-Agent Only-Knowing Revisited , 2010, AlgoSyn.

[14]  Arild Waaler,et al.  Semantics for multi-agent only knowing: extended abstract , 2005, TARK.

[15]  Samir Guglani.  Knowledge , 2016, The Lancet.

[16]  Gerhard Lakemeyer,et al.  Semantical considerations on multiagent only knowing , 2015, Artif. Intell..

[17]  P. T. Geach,et al.  KNOWLEDGE AND BELIEF: An Introduction to the Logic of the Two Notions , 1963 .

[18]  J. Barwise,et al.  Scenes and other Situations , 1981 .

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

[20]  Ronald Fagin,et al.  A model-theoretic analysis of knowledge , 1991, JACM.

[21]  Hector J. Levesque,et al.  All I Know: A Study in Autoepistemic Logic , 1990, Artif. Intell..

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

[23]  Joseph Y. Halpern.  Reasoning about only Knowing with Many Agents , 1993, AAAI.

[24]  Max J. Cresswell,et al.  A New Introduction to Modal Logic , 1998 .

[25]  Gerhard Lakemeyer.  All They Know: A Study in Multi-Agent Autoepistemic Reasoning , 1993, IJCAI.

[26]  Gerhard Lakemeyer,et al.  The logic of knowledge bases , 2000 .

[27]  Jelle Gerbrandy,et al.  Reasoning about Information Change , 1997, J. Log. Lang. Inf..

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