A tale of two diagnoses in probabilistic systems

Diagnosability of partially observable stochastic systems, i.e. the existence of a diagnoser, may be specified in different ways: exact diagnosability requires that faults are almost surely detected and that no fault is erroneously claimed; approximate diagnosability tolerates a small error probability when claiming a fault; last, accurate approximate diagnosability guarantees an arbitrarily small error probability. In this article, we first refine the specification of diagnosability by identifying three criteria: (1) detecting faulty runs or providing information for all runs (2) considering finite or infinite runs, and (3) requiring or not a uniform detection delay. We then give a complete picture of relations between the diagnosability specifications and establish characterisations for most of them. Based on these characterisations, we develop decision procedures, study their complexity and prove their optimality. We also design synthesis algorithms for diagnosers and analyse their memory requirements. Finally we establish undecidability of the diagnosability problems for which we provided no characterisation.