  Complete axiomatizations and exponential-time decision procedures are
provided for reasoning about knowledge and common knowledge when there are
infinitely many agents. The results show that reasoning about knowledge and
common knowledge with infinitely many agents is no harder than when there are
finitely many agents, provided that we can check the cardinality of certain set
differences G - G', where G and G' are sets of agents. Since our complexity
results are independent of the cardinality of the sets G involved, they
represent improvements over the previous results even with the sets of agents
involved are finite. Moreover, our results make clear the extent to which
issues of complexity and completeness depend on how the sets of agents involved
are represented.
