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International Conference on Formal Concept Analysis

ICFCA 2019: Formal Concept Analysis pp 110–129Cite as

Joining Implications in Formal Contexts and Inductive Learning in a Horn Description Logic

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11511))

Abstract

A joining implication is a restricted form of an implication where it is explicitly specified which attributes may occur in the premise and in the conclusion, respectively. A technique for sound and complete axiomatization of joining implications valid in a given formal context is provided. In particular, a canonical base for the joining implications valid in a given formal context is proposed, which enjoys the property of being of minimal cardinality among all such bases. Background knowledge in form of a set of valid joining implications can be incorporated. Furthermore, an application to inductive learning in a Horn description logic is proposed, that is, a procedure for sound and complete axiomatization of \({\mathsf {Horn\text {-}}}\mathcal {M} \) concept inclusions from a given interpretation is developed. A complexity analysis shows that this procedure runs in deterministic exponential time.

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Notes

  1. 1.

    We have not introduced the notion of a concept lattice here, since it is not needed for our purposes; the interested reader is rather referred to [10].

  2. 2.

    The result has not been obtained by hand, but instead the implementation of the algorithm NextClosures [17] in ConceptExplorer FX [16] has been utilized. Thus, no intermediate computation steps are provided.

  3. 3.

    Formally, the role depth is recursively defined as follows: , and , and , and .

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Acknowledgments

The author gratefully thanks Sebastian Rudolph for the very idea of learning in Horn description logics as well as for a helpful discussion on basics of Horn description logics. The author further thanks the reviewers for their constructive remarks.

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Authors and Affiliations

  1. Institute of Theoretical Computer Science, Technische Universität Dresden, Dresden, Germany

    Francesco Kriegel

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  1. Francesco Kriegel
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Correspondence to Francesco Kriegel .

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Editors and Affiliations

  1. Babeș-Bolyai University, Cluj-Napoca, Romania

    Diana Cristea

  2. University of Strasbourg, ENGEES, Strasbourg, France

    Florence Le Ber

  3. Frankfurt University of Applied Sciences, Frankfurt, Germany

    Baris Sertkaya

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Kriegel, F. (2019). Joining Implications in Formal Contexts and Inductive Learning in a Horn Description Logic. In: Cristea, D., Le Ber, F., Sertkaya, B. (eds) Formal Concept Analysis. ICFCA 2019. Lecture Notes in Computer Science(), vol 11511. Springer, Cham. https://doi.org/10.1007/978-3-030-21462-3_9

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  • DOI: https://doi.org/10.1007/978-3-030-21462-3_9

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