On detectability of labeled Petri nets with inhibitor arcs

Detectability is a basic property of dynamic systems: when it holds one can use the observed output signal produced by a system to reconstruct its current state. In this paper, we consider properties of this type in the framework of discrete event systems modeled by Petri nets (a.k.a. place/transition nets). We first study weak detectability and weak approximate detectability. The former implies that there exists an evolution of the net such that all corresponding observed output sequences longer than a given value allow one to reconstruct the current marking (i.e., state). The latter implies that there exists an evolution of the net such that all corresponding observed output sequences longer than a given value allow one to determine if the current marking belongs to a given set. We show that the problem of verifying the first property for labeled place/transition nets with inhibitor arcs and the problem of verifying the second property for labeled place/transition nets are both undecidable. We also consider a property called instant strong detectability which implies that for all possible evolutions the corresponding observed output sequence allows one to reconstruct the current marking. We show that the problem of verifying this property for labeled place/transition nets is decidable while its inverse problem is EXPSPACE-hard.