  Checking whether two pushdown automata with restricted silent actions are
weakly bisimilar was shown decidable by S\'enizergues (1998, 2005). We provide
the first known complexity upper bound for this famous problem, in the
equivalent setting of first-order grammars. This ACKERMANN upper bound is
optimal, and we also show that strong bisimilarity is primitive-recursive when
the number of states of the automata is fixed.
