  In this paper, a general framework is proposed for the analysis and
characterization of observability and diagnosability of finite state systems.
Observability corresponds to the reconstruction of the system's discrete state,
while diagnosability corresponds to the possibility of determining the past
occurrence of some particular states, for example faulty states. A unifying
framework is proposed where observability and diagnosability properties are
defined with respect to a critical set, i.e. a set of discrete states
representing a set of faults, or more generally a set of interest. These
properties are characterized and the involved conditions provide an estimation
of the delay required for the detection of a critical state, of the precision
of the delay estimation and of the duration of a possible initial transient
where the diagnosis is not possible or not required. Our framework makes it
possible to precisely compare some of the observability and diagnosability
notions existing in the literature with the ones introduced in our paper, and
this comparison is presented.
