Observability properties of Petri nets

We discuss the problem of estimating the marking of a place/transition net based on event observation. We assume that the net structure is known while the initial marking is unknown. We define several observability properties and show how they can be proved. In particular, we set up a hierarchy considering the possibility that the above properties are satisfied by a net N starting from an initial marking M₀, by a net N starting from any initial marking M reachable from an initial marking M₀, or by a net N starting from any marking in N^m, where m is the number of places of the net