Verifying one hundred prisoners and a lightbulb

Abstract

This is a case-study in knowledge representation and dynamic epistemic protocol verification. We analyze the ‘one hundred prisoners and a lightbulb’ puzzle. In this puzzle it is relevant what the agents (prisoners) know, how their knowledge changes due to observations, and how they affect the state of the world by changing facts, i.e., by their actions. These actions depend on the history of previous actions and observations. Part of its interest is that all actions are local, i.e. not publicly observable, and part of the problem is therefore how to disseminate local results to other agents, and make them global. The various solutions to the puzzle are presented as protocols (iterated functions from agent's local states, and histories of actions, to actions). The paper consists of three parts. First, we present different versions of the puzzle, and their solutions. This includes a probabilistic version, and a version assuming synchronicity (the interval between prisoners' interrogations is known). The latter is very informative for the prisoners, and allows different protocols (with faster expected termination). Then, we model the puzzle in an epistemic logic incorporating dynamic operators for the effects of information changing events. Such events include both informative actions, where agents become more informed about the non-changing state of the world, and factual changes, wherein the world and the facts describing it change themselves as well. Finally, we verify the basic protocol to solve the problem. Novel contributions in this paper are: Firstly, Protocol 2 and Protocol 4. Secondly, the modelling in dynamic epistemic logic in its entirety — we do not know of a case study that combines factual and informational dynamics in a setting of non-public events, or of a similar proposal to handle asynchronous behaviour in a dynamic epistemic logic. Thirdly, our method to verify dynamic epistemic protocols by reasoning over possibly infinite execution sequences of these protocols. A precursor of the present paper, entitled ‘One hundred prisoners and a lightbulb – logic and computation’ (van Ditmarsch et al., 2010), was presented at KR 2010, Toronto. The differences with the present contribution are as follows: the former contains a section with computational results (expected runtime of different protocols before termination), whereas the focus of the present paper is the verification of one of the presented protocols in the former.