  We consider two-player games over graphs and give tight bounds on the memory
size of strategies ensuring safety objectives. More specifically, we show that
the minimal number of memory states of a strategy ensuring a safety objective
is given by the size of the maximal antichain of left quotients with respect to
language inclusion. This result holds for all safety objectives without any
regularity assumptions. We give several applications of this general principle.
In particular, we characterize the exact memory requirements for the opponent
in generalized reachability games, and we prove the existence of positional
strategies in games with counters.
