Decentralized diagnosability of regular languages is undecidable

We study decentralized fault diagnosis in the context of discrete-event systems. The objective is for a group of agents (the diagnosers) to determine whether a plant has generated a faulty behavior or not, and this, a bounded number of steps after the fault occurred. The plant is modeled as a finite-state automaton producing sequences of events, over some alphabet. One or more special events model faults. Each diagnoser observes only a subset of events generated by the plant. The diagnosers can communicate their observations, which are delivered without loss and in order, but with all arbitrary delay. Diagnosability is a property of the plant with respect to one or more observable event sets, which ensures the existence of diagnosers. In the centralized case, diagnosability is known to be decidable (in polynomial time). We show that the decentralized fault diagnosis problem is undecidable for two or more diagnosers. We also show that the problem is undecidable for three or more diagnosers, even in the case where the language produced by the plant is prefix-closed. We illustrate the modeling framework through an example of fault diagnosis in a simple wireless network.