Synthesis of Distributed Systems from Knowledge-Based Specifications

We consider the problem of synthesizing protocols in a distributed setting satisfying specifications phrased in the logic of linear time and knowledge. In general, synthesis in distributed settings is undecidable already for lineartime temporal logic specifications, but there exist special cases in which synthesis from linear-time temporal logic specifications is known to be decidable. On the basis of these results and a result on the decidability of synthesis of temporal and knowledge specifications in systems with a single agent, van der Meyden and Vardi [CONCUR 96] conjectured that synthesis of temporal and knowledge specifications would be decidable in two classes of environments: hierarchical environments, in which each agent in a linear sequence observes at least as much as the preceding agents, and broadcast environments, in which all communication is constrained to be by synchronous broadcast. We show that this conjecture is true in the case of broadcast environments, but false in the case of even a very simple type of hierarchical environment, where only two agents are involved, one of which observes every aspect of the system state and one of which observes nothing of it. Nevertheless, synthesis from linear-time logic specifications is decidable in hierarchical environments. Moreover, for specifications that are positive in the knowledge modalities, the synthesis problem can be reduced to the same problem for the logic of linear time. We use these facts to conclude the decidability in hierarchical systems of a property closely related to nondeducibility on strategies, a notion that has been studied in computer security.