  We study \emph{partial-information} two-player turn-based games on graphs
with omega-regular objectives, when the partial-information player has
\emph{limited memory}. Such games are a natural formalization for reactive
synthesis when the environment player is not genuinely adversarial to the
system player. The environment player has goals of its own, but the exact goal
of the environment player is unknown to the system player. We prove that the
problem of determining the existence of a winning strategy for the system
player is PSPACE-hard for reachability, safety, and parity objectives.
Moreover, when the environment player is memoryless, the problem is
PSPACE-complete. However, it is simpler to decide if the environment player has
a winning strategy; it is only NP-complete. Additionally, we construct a game
where the the partial-information player needs at least $\mathcal{O}(\sqrt{n})$
bits of memory to retain winning strategies in a game of size $\mathcal{O}(n)$.
