Opacity Issues in Games with Imperfect Information

We study in depth the class of games with opacity condition, which are two-player games with imperfect information in which one of the players only has imperfect information, and where the winning condition relies on the information he has along the play. Those games are relevant for security aspects of computing systems: a play is opaque whenever the player who has imperfect information never “knows” for sure that the current position is one of the distinguished “secret” positions. We study the problems of deciding the existence of a winning strategy for each player, and we call them the opacity-violate problem and the opacity-guarantee problem. Focusing on the player with perfect information is new in the field of games with imperfect-information because when considering classical winning conditions it amounts to solving the underlying perfect-information game. We establish the EXPTIME-completeness of both above-mentioned problems, showing that our winning condition brings a gap of complexity for the player with perfect information, and we exhibit the relevant opacity-verify problem, which noticeably generalizes approaches considered in the literature for opacity analysis in discrete-event systems. In the case of blindfold games, this problem relates to the two initial ones, yielding the determinacy of blindfold games with opacity condition and the PSPACE-completeness of the three problems.