Verification and enforcement of current-state opacity based on a state space approach

Abstract

In this study, the current-state opacity verification and enforcement problems in the context of partially-observed discrete event systems are investigated based on an algebraic state space approach. Current-state opacity is defined as a measure of whether a system can protect current vital information from exposure. If given systems are not current-state opaque, it is necessary to synthesize a supervisor to restrict systems’ behavior and enforce the property. In this paper, the algebraic-state-space-based approach to these problems provides a novel perspective to the literature. Firstly, an equivalent algebraic structure of a partially-observed finite automaton is established with the help of the semi-tensor product, where the complexity of an observable structure matrix is O(|X|2|Eo|) (X is the set of states and Eo is the set of observable events). And then, using the breadth-first search method, an algorithm to ascertain the state estimate is proposed based on the observable structure matrix, the complexity of which is O(|X|3|Eo|). For the verification of current-state opacity, it is carried out by determining whether or not there exists a leaking state estimate in the set of secret-related state estimates. In the worst case, this process would be done in O(2|X|). For the enforcement of opacity, the algebraic expression of a controlled system is also constructed, under which, the breadth-first search approach can still be applied to ascertain state estimates under each control profile. How to obtain all permissive control profiles is summarized as Algorithm 2, the complexity of which is O(2|E|×|X|3×|Eo|+2|X|).