Critical Observability for Automata and Petri Nets

Critical observability is a property of cyber-physical systems to detect whether the current state belongs to a set of critical states. In safety-critical applications, critical states model operations that may be unsafe or of a particular interest. De Santis et al. introduced critical observability for linear switching systems, and Pola et al. adapted it for discrete-event systems, focusing on algorithmic complexity. We study the computational complexity of deciding critical observability for systems modeled as (networks of) finite-state automata and Petri nets. We show that deciding critical observability is (i) NL-complete for finite automata, that is, it is efficiently verifiable on parallel computers, (ii) PSPACE-complete for networks of finite automata, that is, it is very unlikely solvable in polynomial time, and (iii) undecidable for labeled Petri nets, but becoming decidable if the set of critical states (markings) is finite or co-finite, in which case the problem is as hard as the non-reachability problem for Petri nets.