Synthesis of Opacity-Enforcing Winning Strategies Against Colluded Opponent

This paper studies a language-based opacity enforcement in a two-player, zero-sum game on a graph. In this game, player 1 (P1) wins if it can achieve a secret temporal goal described by the language of a finite automaton, no matter what strategy the opponent player 2 (P2) selects. In addition, P1 aims to win while making its goal opaque to a passive observer with imperfect information. However, P2 colludes with the observer to reveal P1's secret whenever P2 cannot prevent P1 from achieving its goal, and therefore, opacity must be enforced against P2. We show that a winning and opacity-enforcing strategy for P1 can be computed by reducing the problem to solving a reachability game augmented with observer's belief states. Furthermore, if such a strategy does not exist, winning for P1 must entail the price of revealing his secret to the observer. We demonstrate our game-theoretic solution of opacity-enforcement control through a small illustrative example and in a robot motion planning problem.