Online verification of [formula omitted]-step opacity by Petri nets in centralized and decentralized structures

Abstract

This paper presents an online approach to deal with the problem of state-based K-step opacity of a discrete event system modeled with labeled Petri nets. The secret states can be described by a set of either markings or generalized mutual exclusion constraints (GMECs), which is more general than the existing approaches in which the secret states are characterized by GMECs only. We formulate an integer linear programming (ILP) model to find a marking outside the set of secret states from each of the K-delayed state estimates, avoiding a complete enumeration of reachability set. Based on the ILP model, an online algorithm is developed to verify K-step opacity of an observation (i.e., an observed sequence). As an extension of K-step opacity, we define the notion of jointK-step opacity in a decentralized structure with two local sites and extend the proposed algorithm to verify the joint K-step opacity of an observation. In addition, the relationship between joint K-step opacity and K-step opacity is formally discussed.