  We study detectability properties for labeled Petri nets and finite automata.
We first study weak approximate detectability (WAD) that implies that there
exists an infinite observed output sequence of the system such that each prefix
of the output sequence with length greater than a given value allows an
observer to determine if the current state belongs to a given set. We also
consider two new concepts called instant strong detectability (ISD) and
eventual strong detectability (ESD). The former property implies that for each
possible infinite observed output sequence each prefix of the output sequence
allows reconstructing the current state. The latter implies that for each
possible infinite observed output sequence, there exists a value such that each
prefix of the output sequence with length greater than that value allows
reconstructing the current state.
  Results: WAD: undecidable for labeled Petri nets, PSPACE-complete for finite
automata ISD: decidable and EXPSPACE-hard for labeled Petri nets, belongs to P
for finite automata ESD: decidable under promptness assumption and
EXPSPACE-hard for labeled Petri nets, belongs to P for finite automata SD:
belongs to P for finite automata, strengthens Shu and Lin's 2011 results based
on two assumptions of deadlock-freeness and promptness ISD<SD<ESD<WD<WAD for
both labeled Petri nets and finite automata
