We show that strict deterministic propositional dynamic logic with
intersection is highly undecidable, solving a problem in the Stanford
Encyclopedia of Philosophy. In fact we show something quite a bit stronger. We
introduce the construction of program equivalence, which returns the value
T precisely when two given programs are equivalent on halting
computations. We show that virtually any variant of propositional dynamic logic
has Π11​-hard validity problem if it can express even just the equivalence
of well-structured programs with the empty program \texttt{skip}. We also show,
in these cases, that the set of propositional statements valid over finite
models is not recursively enumerable, so there is not even an axiomatisation
for finitely valid propositions.Comment: 8 page