  We study observation-based strategies for two-player turn-based games on
graphs with omega-regular objectives. An observation-based strategy relies on
imperfect information about the history of a play, namely, on the past sequence
of observations. Such games occur in the synthesis of a controller that does
not see the private state of the plant. Our main results are twofold. First, we
give a fixed-point algorithm for computing the set of states from which a
player can win with a deterministic observation-based strategy for any
omega-regular objective. The fixed point is computed in the lattice of
antichains of state sets. This algorithm has the advantages of being directed
by the objective and of avoiding an explicit subset construction on the game
graph. Second, we give an algorithm for computing the set of states from which
a player can win with probability 1 with a randomized observation-based
strategy for a Buechi objective. This set is of interest because in the absence
of perfect information, randomized strategies are more powerful than
deterministic ones. We show that our algorithms are optimal by proving matching
lower bounds.
