This paper is concerned with the synthesis of strategies in network systems with active cyber deception. Active deception in a network employs decoy systems and other defenses to conduct defensive planning against the intrusion of malicious attackers who have been confirmed by sensing systems. In this setting, the defenders objective is to ensure the satisfaction of security properties specified in temporal logic formulas. We formulate the problem of deceptive planning with decoy systems and other defenses as a two-player games with asymmetrical information and Boolean payoffs in temporal logic. We use level-2 hypergame with temporal logic objectives to capture the incomplete/incorrect knowledge of the attacker about the network system as a payoff misperception. The true payoff function is private information of the defender. Then, we extend the solution concepts of $omega$-regular games to analyze the attackers rational strategy given her incomplete information. By generalizing the solution of level-2 hypergame in the normal form to extensive form, we extend the solutions of games with safe temporal logic objectives to decide whether the defender can ensure security properties to be satisfied with probability one, given any possible strategy that is perceived to be rational by the attacker. Further, we use the solution of games with co-safe (reachability) temporal logic objectives to determine whether the defender can engage the attacker, by directing the attacker to a high-fidelity honeypot. The effectiveness of the proposed synthesis methods is illustrated with synthetic network systems with honeypots.
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