Well-structured program equivalence is highly undecidable

Abstract

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 ⊤ precisely when two given programs are equivalent on halting computations. We show that virtually any variant of propositional dynamic logic has a Π 1 1 -hard validity problem if it can express even just the equivalence of well-structured programs with the empty program 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 axiomatization for finitely valid propositions.