Two Ways to Common Knowledge

It is not clear what a system for evidence-based common knowledge should look like if common knowledge is treated as a greatest fixed point. This paper is a preliminary step towards such a system. We argue that the standard induction rule is not well suited to axiomatize evidence-based common knowledge. As an alternative, we study two different deductive systems for the logic of common knowledge. The first system makes use of an induction axiom whereas the second one is based on co-inductive proof theory. We show the soundness and completeness for both systems.

[1]  Thomas Studer,et al.  On the Proof Theory of the Modal mu-Calculus , 2008, Stud Logica.

[2]  Melvin Fitting,et al.  The logic of proofs, semantically , 2005, Ann. Pure Appl. Log..

[3]  David Walker,et al.  Local Model Checking in the Modal mu-Calculus , 1991, Theor. Comput. Sci..

[4]  Martin Hofmann,et al.  A Proof System for the Linear Time µ-Calculus , 2006, FSTTCS.

[5]  Sergei N. Artëmov.  Explicit provability and constructive semantics , 2001, Bull. Symb. Log..

[6]  W. Browder,et al.  Annals of Mathematics , 1889 .

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

[8]  Naveen Garg,et al.  FSTTCS 2006: Foundations of Software Technology and Theoretical Computer Science, 26th International Conference, Kolkata, India, December 13-15, 2006, Proceedings , 2006, FSTTCS.

[9]  I. Walukiewicz.  Games for the -calculus , 2007 .

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

[11]  Igor Walukiewicz,et al.  Games for the mu-Calculus , 1996, Theor. Comput. Sci..

[12]  S. Shelah,et al.  Annals of Pure and Applied Logic , 1991 .

[13]  E. Allen Emerson,et al.  An Automata Theoretic Decision Procedure for the Propositional Mu-Calculus , 1989, Inf. Comput..

[14]  Gerhard Jäger,et al.  About cut elimination for logics of common knowledge , 2005, Ann. Pure Appl. Log..

[15]  Sergei N. Artëmov.  Justified common knowledge , 2006, Theor. Comput. Sci..

[16]  Colin Stirling,et al.  Modal Mu-Calculi , 2001 .

[17]  Sergei N. Artëmov.  THE LOGIC OF JUSTIFICATION , 2008, The Review of Symbolic Logic.

[18]  Colin Stirling,et al.  Modal mu-calculi , 2007, Handbook of Modal Logic.

[19]  Sergei N. Artëmov,et al.  Logical omniscience as a computational complexity problem , 2009, TARK '09.

[20]  Sergei N. Artëmov.  Logic of Proofs , 1994, Ann. Pure Appl. Log..

[21]  Sergei N. Artemovy.  Operational Modal Logic , 1995 .

[22]  Evangelia Antonakos.  Justified and Common Knowledge: Limited Conservativity , 2007, LFCS.

[23]  Roman Kuznets,et al.  Self-Referential Justifications in Epistemic Logic , 2010, Theory of Computing Systems.