The Complexity of Games on Higher Order Pushdown Automata

We prove an n-EXPTIME lower bound for the problem of deciding the winner in a reachability game on Higher Order Pushdown Automata (HPDA) of level n. This bound matches the known upper bound for parity games on HPDA. As a consequence the mu-calculus model checking over graphs given by n-HPDA is n-EXPTIME complete