Beyond: Exploring Non-Regular Extensions of PDL with Description Logics Features

Bartosz Bednarczyk

Abstract: We investigate the impact of non-regular path expressions on the decidability of satisfiability checking and querying in description logics. Our primary object of interest is \(\mathcal {ALC}_{\textsf {vpl}}\), an extension of \(\mathcal {ALC}\) with path expressions using visibly-pushdown languages, which was shown to be decidable by Löding et al. in 2007. The paper present a series of undecidability results. We prove undecidability of \(\mathcal {ALC}_{\textsf {vpl}}\) with the seemingly innocent \(\textsf{Self}\) operator. Then, we consider the simplest non-regular (visibly-pushdown) language \( r ^\# s ^\# {:}{=}\{ r ^n s ^n \mid n \in \mathbb {N}\}\). We establish undecidability of the concept satisfiability problem for \({\mathcal {A}\mathcal {L}\mathcal {C}}_{\textsf {reg}}\) extended with nominals and \( r ^\# s ^\#\), as well as of the query entailment problem for \(\mathcal {ALC}\)-TBoxes, where such non-regular atoms are present in queries.