Reasoning About Knowledge is the first book to provide a general discussion of approaches to reasoning about knowledge and its applications to distributed ...

This paper studies the role that known bounds on message transmission times in a computer network play on the evolution of the epistemic state over time. A connection to cones of causal influence analogous to, and more general than, light cones is presented. Focusing on lower bounds on message transmission times, an analysis is presented of how knowledge about when others are guaranteed to be ignorant about an event of interest can arise. This has implications in competitive settings, in which (...) knowing about another’s ignorance can provide an advantage. (shrink)

The effect of upper bounds on message delivery times in a computer network upon the dynamics of knowledge gain is investigated. Recent work has identified centipedes and brooms—causal structures that combine message chains with time bound information—as necessary conditions for knowledge gain and common knowledge gain, respectively. This paper shows that, under the full-information protocol, these structures are both necessary and sufficient for such epistemic gain. We then apply this analysis to gain insights into the relation between “everyone knows” and (...) common knowledge. We prove a tight threshold on the depth k, beyond which Ek G (everyone in G knows nested to depth k) collapses into CG (common knowledge), when this knowledge concerns the occurrence of a spontaneous event. The threshold depends on the size of the group G of agents, as well as the time that has elapsed since the event of interest occurred. The existence of such a threshold is not guaranteed for all protocols, which is demonstrated here by presenting a counterexample in which no such threshold exists. (shrink)

A teacher announced to his pupils that on exactly one of the days of the following school week (Monday through Friday) he would give them a test. But it would be a surprise test; on the evening before the test they would not know that the test would take place the next day. One of the brighter students in the class then argued that the teacher could never give them the test. "It can't be Friday," she said, "since in that (...) case we'll expect it on Thurday evening. But then it can't be Thursday, since having already eliminated Friday we'll know Wednesday evening that it has to be Thursday. And by similar reasoning we can also eliminate Wednesday, Tuesday, and Monday. So there can't be a test!" The students were somewhat baffled by the situation. The teacher was well-known to be truthful, so if he said there would be a test, then it was safe to assume that there would be one. On the other hand, he also said that the test would be a surprise. But it seemed that whenever he gave the test, it wouldn't be a surprise. Well, the teacher gave the test on Tuesday, and, sure enough, the students were surprised. (shrink)