We study networks of processes that all execute the same finite state protocol and that communicate through broadcasts. The processes are organized in a graph (a topology) and only the neighbors of a process in this graph can receive its broadcasts. The coverability problem asks, given a protocol and a state of the protocol, whether there is a topology for the processes such that one of them (at least) reaches the given state. This problem is undecidable. We study here an under-approximation of the problem where processes alternate a bounded number of times $k$ between phases of broadcasting and phases of receiving messages. We show that, if the problem remains undecidable when $k$ is greater than 6, it becomes decidable for $k=2$, and EXPSPACE-complete for $k=1$. Furthermore, we show that if we restrict ourselves to line topologies, the problem is in $P$ for $k=1$ and $k=2$.

翻译：我们研究由执行相同有限状态协议并通过广播进行通信的进程所构成的网络。这些进程被组织在一个图（拓扑结构）中，且只有图中某个进程的邻居才能接收其广播。覆盖性问题要求判断：对于给定的协议及协议状态，是否存在一种进程拓扑结构，使得至少一个进程能够到达给定状态。该问题是不可判定的。本文研究该问题的一个下近似，其中进程在广播阶段和接收消息阶段之间交替执行有限次数 $k$。我们证明，当 $k$ 大于 6 时问题仍不可判定，但在 $k=2$ 时变为可判定，在 $k=1$ 时为 EXPSPACE 完全问题。此外，我们证明若将拓扑结构限制为线型，则问题在 $k=1$ 和 $k=2$ 时均属于 $P$ 类。