Verification and enforcement of strong infinite- and [formula omitted]-step opacity using state recognizers

Abstract

In this paper, we study the verification and enforcement problems of strong infinite-step opacity and k-step opacity for partially observed discrete-event systems modeled by finite state automata. Strong infinite-step opacity is a property such that the visit of a secret state cannot be inferred by an intruder at any instance along the entire observation trajectory, while strong k-step opacity is a property such that the visit of a secret state cannot be inferred within k steps after the visit. We propose two information structures called an ∞-step recognizer and a k-step recognizer to verify these two properties. The complexities of our algorithms to verify strong infinite- and k-step opacity are O(22⋅|X|⋅|Eo|) and O(2(k+2)⋅|X|⋅|Eo|), respectively, which are lower than that of existing methods in the literature (|X| and |Eo| are the numbers of states and observable events in a plant, respectively). We also derive an upper bound for the value of k in strong k-step opacity, and propose an effective algorithm to determine the maximal value of k for a given plant. Finally, we note that enforcement of strong infinite- and k-step opacity can be transformed into a language specification enforcement problem and hence be solved using supervisory control.