Levels of Knowledge in Distributed Computing 1 Rohit Parikh 2, 3, 4 and Paul Krasucki 2,3 Abstract: We investigate the properties of the state of knowledge of a system of a processes taking part in a distributed computation. In particular we show a 1-1 correspondence between levels of knowledge and certain regular sets of strings on the alphabet {K 1 , ..., K n } where 1, ..., n are processes and K i stands for “i knows that” Introduction: You and I approach an intersection. You have the right of way. You should go, I should stop; that is the law and we both know it. However, that is not enough. You should know that I know. You do not want to risk your life merely to assert your right of way. Also, I should know that you know I am going to stop. If not, you will stop (even though you have the right of way) and neither of us will go; we do not want a deadlock. David Lewis in [L] and Clark and Marshall [CM2] argue that the condition of common knowledge is necessary for such co-ordinated actions. For another example of the relevance of common knowledge, it is proved in [HM] that clock synchronisation is impossible without common knowledge. Yet another example that we have not seen in the literature but is doubtless discussed somewhere is that mutual deterrence between the superpowers also depends on common knowledge. Other procedures need less than common knowledge. Suppose there are three dirty children, each of whom can see the foreheads of the others but not his/her own. A parent arrives on the scene and announces “One of you has a dirty forehead”. Then the children are asked, “Is your forehead dirty?”. The first two children say “I don’t know”, but the third child says “My forehead is dirty”. It can be shown that common knowledge is not necessary for this inference, but it is enough if the first child knows D (someone has a dirty forehead), the second child knows that the first knows D, and the third knows that the second knows that the first knows D. Thus, even though the parent’s announcement creates common knowledge, it is not really needed. Sometimes it may even be undesirable to have common knowledge. Suppose Mr. A is having an affair with Mrs. B. Of course they do not want Mrs. A or Mr. B to know. On the other hand, if Mrs. A does find out, she may wish to hide her knowledge to avoid a violent confrontaion and the breakup of two families. It is clear that groups of people can have knowledge of the world and each other’s knowledge in various complicated ways. What are all the possibilities? 1 A preliminary version of this paper appeared in Proc. IEEE Symposium on Logic in Computer Science, Boston 1986, pp.314-321 2 Department of Computer Science, Brooklyn College of CUNY, Research supported by NSF grant DCR 85-04825 and a CUNY FRAP Grant. 3 Department of Computer Science, CUNY Graduate Center 4 Mathematics Department, CUNY Graduate Center, 33 West 42nd Street, New York, NY, 10036 1