Summary: We consider two-player games over graphs and give tight bounds on the memory size of strategies ensuring safety conditions. More specifically, we show that the minimal number of memory states of a strategy ensuring a safety condition is given by the size of the maximal antichain of left quotients with respect to language inclusion. This result holds for all safety conditions without any regularity assumptions, and for all (finite or infinite) graphs of finite degree.
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.
For the entire collection see [Zbl 1329.68036].