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Beyond \(\mathcal {ALC}_{\textsf {reg}}\): Exploring Non-Regular Extensions of PDL with Description Logics Features

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Logics in Artificial Intelligence (JELIA 2023)

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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.

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Acknowledgements

This work was supported by the ERC Consolidator Grant No. 771779 (DeciGUT). Snake and cobra icons were downloaded from Icons8 and Flaticon.

The author would like to thank Reijo Jaakkola for many inspiring discussions; Sebastian Rudolph for pinpointing [9] and suggesting many improvements, especially related to the previous versions of Sect. 4; Witold Charatonik for very careful proofreading and his pedantic approach to writing; as well as Alessio Mansutti and Emanuel Kieroński for their help polishing the introduction.

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Authors and Affiliations

  1. Computational Logic Group, Technische Universität Dresden, Dresden, Germany

    Bartosz Bednarczyk

  2. Institute of Computer Science, University of Wrocław, Wrocław, Poland

    Bartosz Bednarczyk

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  1. Bartosz Bednarczyk
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Correspondence to Bartosz Bednarczyk .

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Editors and Affiliations

  1. TU Dresden, Dresden, Germany

    Sarah Gaggl

  2. Artificial Intelligence Research Institute - IIIA CSIC, Barcelona, Spain

    Maria Vanina Martinez

  3. Umeå University, Umeå, Sweden

    Magdalena Ortiz

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Bednarczyk, B. (2023). Beyond \(\mathcal {ALC}_{\textsf {reg}}\): Exploring Non-Regular Extensions of PDL with Description Logics Features. In: Gaggl, S., Martinez, M.V., Ortiz, M. (eds) Logics in Artificial Intelligence. JELIA 2023. Lecture Notes in Computer Science(), vol 14281. Springer, Cham. https://doi.org/10.1007/978-3-031-43619-2_21

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  • DOI: https://doi.org/10.1007/978-3-031-43619-2_21

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