Enforcing opacity with publicly known edit functions under incomparable observations

This paper investigates opacity enforcement via publicly known and constrained edit functions under incomparable observations. We first formalize the notion of (Formula presented) -enforceability, which combines admissibility, consistency, confidentiality, and integrity requirements. A game-theoretic synthesis framework is then developed, consisting of three pruning stages and one merging stage, including: (i) an edit game structure is constructed to capture all feasible constrained edit actions; (ii) problematic states that violate admissibility, confidentiality, or consistency are pruned; (iii) an identifying observer is employed to model the reverse-engineering capability of the intruder; and (iv) states are merged to ensure the consistency of edit actions under the observation of the defender. The resulting edit mechanism provides necessary and sufficient conditions for synthesizing (Formula presented) -enforcing edit functions.