  We give a direct polynomial-time reduction from parity games played over the
configuration graphs of collapsible pushdown systems to safety games played
over the same class of graphs. That a polynomial-time reduction would exist was
known since both problems are complete for the same complexity class. Coming up
with a direct reduction, however, has been an open problem. Our solution to the
puzzle brings together a number of techniques for pushdown games and adds three
new ones. This work contributes to a recent trend of liveness to safety
reductions which allow the advanced state-of-the-art in safety checking to be
used for more expressive specifications.
