Abstract:

Complete axiomatizations and exponential-time decision procedures are provided for reasoning about knowledge (i.e. statements of the form that every agent in a group G of agents knows that p and also that every agent in group G' knows q) and common knowledge (statements that express the idea that every agent in group G knows p and that every agent in G knows that every agent in G knows p, etc.) 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. The work to be discussed is joint with Joe Halpern (Cornell, CS).