  Parse trees are fundamental syntactic structures in both computational
linguistics and compilers construction. We argue in this paper that, in both
fields, there are good incentives for model-checking sets of parse trees for
some word according to a context-free grammar. We put forward the adequacy of
propositional dynamic logic (PDL) on trees in these applications, and study as
a sanity check the complexity of the corresponding model-checking problem:
although complete for exponential time in the general case, we find natural
restrictions on grammars for our applications and establish complexities
ranging from nondeterministic polynomial time to polynomial space in the
relevant cases.
