In this chapter we look at how one might represent statements such as “John knows that it is raining,” “John believes that it will rain tomorrow,” “Mary knows that John believes that it will rain tomorrow” and “It is common knowledge between Mary and John that it is raining.”

The partition model of knowledge

Consider a distributed system, in which multiple processors autonomously performing some joint computation. Of course, the joint nature of the computation means that the processors need to communicate with one another. One set of problems comes about when the communication is error prone. In this case the system analyst may find himself saying something like the following: “Processor A sent the message to processor B. The message may not arrive, and processor A knows this. Furthermore, this is common knowledge, so processor A knows that processor B knows that it (A) knows that if a message was sent it may not arrive.” The topic of this chapter is how to make such reasoning precise.

Muddy children and warring generals

Often the modeling is done in the context of some stylized problem, with an associated entertaining story. Thus, for example, when we return to the distributed computing application in Section 13.4, rather than speak about computer processors, we will tell the story of two generals who attempt to coordinate among themselves to gang up on a third.