Reasoning About Common Knowledge with Infinitely Many Agents

TL;DR

This paper develops complete axiomatizations and decision procedures for reasoning about knowledge and common knowledge among infinitely many agents, showing that the complexity is comparable to finite cases under certain conditions.

Contribution

It introduces new complexity results and decision procedures for infinite-agent scenarios, extending prior finite-agent frameworks and clarifying the role of set representations.

Findings

Reasoning complexity is similar for finite and infinite agents under certain conditions.

New decision procedures are exponential-time and complete for infinite-agent knowledge logic.

Results are independent of the size of agent sets involved.

Abstract

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.