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Not Too Big, Not Too Small... Complexities of Fixed-Domain Reasoning in First-Order and Description Logics

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Progress in Artificial Intelligence (EPIA 2017)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10423))

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Abstract

We consider reasoning problems in description logics and variants of first-order logic under the fixed-domain semantics, where the model size is finite and explicitly given. It follows from previous results that standard reasoning is NP-complete for a very wide range of logics, if the domain size is given in unary encoding. In this paper, we complete the complexity overview for unary encoding and investigate the effects of binary encoding with partially surprising results. Most notably, fixed-domain standard reasoning becomes NExpTime for the rather low-level description logics \(\mathcal {ELI}\) and \(\mathcal {ELF}\) (as opposed to ExpTime when no domain size is given). On the other hand, fixed-domain reasoning remains NExpTime even for first-order logic, which is undecidable under the unconstrained semantics. For less expressive logics, we establish a generic criterion ensuring NP-completeness of fixed-domain reasoning. Amongst other logics, this criterion captures all the tractable profiles of OWL 2.

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Notes

  1. 1.

    For brevity, we omit the global restrictions of \(\mathcal {SROIQ}\) as they are irrelevant in our setting.

  2. 2.

    Disjointness (\(A \sqsubseteq \lnot B\)) of concepts A, B are modeled in \(\mathcal {ELI}\) as \(A \sqcap B \sqsubseteq C_{\bot }\), where \(C_{\bot }\) is a freshly introduced concept name that acts as the bottom concept in countermodels [1].

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Acknowledgements

This work is supported by DFG in the Research Training Group QuantLA (GRK 1763). We thank Franz Baader for asking the right questions, and are grateful for the valuable feedback from the anonymous reviewers, which helped greatly to improve this work.

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Authors and Affiliations

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

    Sebastian Rudolph & Lukas Schweizer

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  1. Sebastian Rudolph
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  2. Lukas Schweizer
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Correspondence to Lukas Schweizer .

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Editors and Affiliations

  1. Universidade do Porto, Porto, Portugal

    Eugénio Oliveira

  2. Universidade do Porto, Porto, Portugal

    João Gama

  3. Polytechnic Institute of Porto, Porto, Portugal

    Zita Vale

  4. Universidade do Porto, Porto, Portugal

    Henrique Lopes Cardoso

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Rudolph, S., Schweizer, L. (2017). Not Too Big, Not Too Small... Complexities of Fixed-Domain Reasoning in First-Order and Description Logics. In: Oliveira, E., Gama, J., Vale, Z., Lopes Cardoso, H. (eds) Progress in Artificial Intelligence. EPIA 2017. Lecture Notes in Computer Science(), vol 10423. Springer, Cham. https://doi.org/10.1007/978-3-319-65340-2_57

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  • DOI: https://doi.org/10.1007/978-3-319-65340-2_57

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  • Online ISBN: 978-3-319-65340-2

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