  We study the bisimilarity problem for probabilistic pushdown automata (pPDA)
and subclasses thereof. Our definition of pPDA allows both probabilistic and
non-deterministic branching, generalising the classical notion of pushdown
automata (without epsilon-transitions). Our first contribution is a general
construction that reduces checking bisimilarity of probabilistic transition
systems to checking bisimilarity of non-deterministic transition systems. This
construction directly yields decidability of bisimilarity for pPDA, as well as
an elementary upper bound for the bisimilarity problem on the subclass of
probabilistic basic process algebras, i.e., single-state pPDA. We further show
that, with careful analysis, the general reduction can be used to prove an
EXPTIME upper bound for bisimilarity of probabilistic visibly pushdown
automata. Here we also provide a matching lower bound, establishing
EXPTIME-completeness. Finally we prove that deciding bisimilarity of
probabilistic one-counter automata, another subclass of pPDA, is
PSPACE-complete. Here we use a more specialised argument to obtain optimal
complexity bounds.
