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Dualization in Lattices Given by Implicational Bases

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Formal Concept Analysis (ICFCA 2019)

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

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Abstract

It was recently proved that the dualization in lattices given by implicational bases is impossible in output-polynomial time unless \(\mathsf{P}\!=\!\mathsf{NP}\). In this paper, we show that this result holds even when the premises in the implicational base are of size at most two. In the case of premises of size one—when the lattice is distributive—we show that the dualization is possible in output quasi-polynomial time whenever the graph of implications is of bounded maximum induced matching. Lattices that share this property include distributive lattices coded by the ideals of an interval order.

This work has been supported by the ANR project GraphEn ANR-15-CE40-0009.

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Notes

  1. 1.

    Here the graph of implications is considered directed, where there is an arc (ab) in \(G(\varSigma )\) if there exists \(A\rightarrow b\in \varSigma \), \(a\in A\), and the implicational base is said to be acyclic if its graph of implications is; see [22].

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Authors and Affiliations

  1. LIMOS, Université Clermont Auvergne, Aubière, France

    Oscar Defrain & Lhouari Nourine

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  1. Oscar Defrain
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  2. Lhouari Nourine
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Correspondence to Oscar Defrain .

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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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Defrain, O., Nourine, L. (2019). Dualization in Lattices Given by Implicational Bases. 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_7

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  • DOI: https://doi.org/10.1007/978-3-030-21462-3_7

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  • Online ISBN: 978-3-030-21462-3

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