  We provide algorithmically verifiable necessary and sufficient conditions for
fundamental system theoretic properties of discrete time linear systems subject
to data losses. More precisely, the systems in our modeling framework are
subject to disruptions (data losses) in the feedback loop, where the set of
possible data loss sequences is captured by an automaton. As such, the results
are applicable in the context of shared (wireless) communication networks
and/or embedded architectures where some information on the data loss behaviour
is available a priori. We propose an algorithm for deciding observability (or
the absence of it) for such systems, and show how this algorithm can be used
also to decide other properties including constructibility, controllability,
reachability, null-controllability, detectability and stabilizability by means
of relations that we establish among these properties. The main apparatus for
our analysis is the celebrated Skolem Theorem from linear algebra. Moreover, we
study the relation between the model adopted in this paper and a previously
introduced model where, instead of allowing dropouts in the feedback loop, one
allows for time varying delays.
