Common knowledge does not have the Beth property

Common knowledge is an essential notion for coordination among agents. We show that the logic of common knowledge does not have the Beth property and thus it also lacks interpolation. The proof we present is a variant of Maksimova's proof that temporal logics with 'the next' do not have the Beth property. Our result also provides an explanation why it is so difficult to find 'nice' deductive systems for common knowledge.

[1]  Thomas Studer,et al.  Cut-free common knowledge , 2007, J. Appl. Log..

[2]  L. L. Maksimova,et al.  An analog of Beth's theorem in normal extensions of the modal logic K4 , 1992 .

[3]  Lev Gordeev,et al.  Basic proof theory , 1998 .

[4]  D. Gabbay,et al.  Interpolation and Definability: Modal and Intuitionistic Logic , 2005 .

[5]  L. L. Maksimova,et al.  Temporal logics with “the next” operator do not have interpolation or the Beth property , 1991 .

[6]  K. Jon Barwise,et al.  Three Views of Common Knowledge , 1988, TARK.

[7]  D. Gabbay,et al.  Interpolation and Definability in Modal Logics (Oxford Logic Guides) , 2005 .

[8]  Thomas A. Henzinger,et al.  Abstractions from proofs , 2004, POPL.

[9]  Larisa Maksimova,et al.  Temporal logics of "the next" do not have the Beth property , 1999, J. Appl. Non Class. Logics.

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

[11]  Rupak Majumdar,et al.  State of the Union: Type Inference Via Craig Interpolation , 2007, TACAS.

[12]  Thomas Studer,et al.  Syntactic cut-elimination for common knowledge , 2009, Ann. Pure Appl. Log..

[13]  Dov M. Gabbay,et al.  Interpolation in Practical Formal Development , 2001, Log. J. IGPL.

[14]  M. Kracht.  Tools and Techniques in Modal Logic , 1999 .

[15]  Samuel R. Buss,et al.  Chapter I - An Introduction to Proof Theory , 1998 .

[16]  D. Gabbay.  Craig's interpolation theorem for modal logics , 1972 .

[17]  J. Girard.  Proof Theory and Logical Complexity , 1989 .

[18]  Rajeev Goré,et al.  Cut-free Single-pass Tableaux for the Logic of Common Knowledge , 2007 .

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

[20]  S. Buss.  Handbook of proof theory , 1998 .

[21]  William Craig,et al.  Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory , 1957, Journal of Symbolic Logic.

[22]  Sheila A. McIlraith,et al.  Partition-based logical reasoning for first-order and propositional theories , 2005, Artif. Intell..

[23]  Greg Nelson,et al.  Simplification by Cooperating Decision Procedures , 1979, TOPL.

[24]  Javier Esparza,et al.  Abstraction Refinement with Craig Interpolation and Symbolic Pushdown Systems , 2006, J. Satisf. Boolean Model. Comput..

[25]  W. van der Hoek,et al.  Epistemic logic for AI and computer science , 1995, Cambridge tracts in theoretical computer science.