  We investigate the impact of non-regular path expressions on the decidability
of satisfiability checking and querying in description logics extending ALC.
Our primary objects of interest are ALCreg and ALCvpl, the extensions of with
path expressions employing, respectively, regular and visibly-pushdown
languages. The first one, ALCreg, is a notational variant of the well-known
Propositional Dynamic Logic of Fischer and Ladner. The second one, ALCvpl, was
introduced and investigated by Loding and Serre in 2007. The logic ALCvpl
generalises many known decidable non-regular extensions of ALCreg.
  We provide a series of undecidability results. First, we show that
decidability of the concept satisfiability problem for ALCvpl is lost upon
adding the seemingly innocent Self operator. Second, we establish
undecidability for the concept satisfiability problem for ALCvpl extended with
nominals. Interestingly, our undecidability proof relies only on one single
non-regular (visibly-pushdown) language, namely on r#s# := { r^n s^n | n in N }
for fixed role names r and s. Finally, in contrast to the classical database
setting, we establish undecidability of query entailment for queries involving
non-regular atoms from r#s#, already in the case of ALC-TBoxes.
