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Algorithmic Sahlqvist Preservation for Modal Compact Hausdorff Spaces

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Logic, Language, Information, and Computation (WoLLIC 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10388))

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Abstract

In this paper, we use the algorithm ALBA to reformulate the proof in [1, 2] that over modal compact Hausdorff spaces, the validity of Sahlqvist sequents are preserved from open assignments to arbitrary assignments. In particular, we prove an adapted version of the topological Ackermann lemma based on the Esakia-type lemmas proved in [1, 2].

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Notes

  1. 1.

    Notice that the name “frame” occurs in two different ways in the present paper, one is in point-free topology, the other is in modal logic. Here we use the name “Kripke frame” to refer to the notion in modal logic and “frame” to refer to the notion in point-free topology.

  2. 2.

    The condition 3 is well-known in [18].

  3. 3.

    That is, not only finite meets and complete joins are preserved, but also the modal operators, i.e. \(\Box _{\mathbb {B}_{\mathbb {F}}} X=\Box _{\mathcal {T}} X\) and \((\Box _{\mathbb {B}_{\mathbb {F}}} X^c)^c=\Diamond _{\mathcal {T}} X\) for all \(X\in \tau \).

  4. 4.

    For these terminologies, see [10].

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Author information

Authors and Affiliations

  1. Delft University of Technology, Delft, The Netherlands

    Zhiguang Zhao

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  1. Zhiguang Zhao
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Correspondence to Zhiguang Zhao .

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Editors and Affiliations

  1. Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland

    Juliette Kennedy

  2. Centro de Informática, Recife, Pernambuco, Brazil

    Ruy J.G.B. de Queiroz

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Zhao, Z. (2017). Algorithmic Sahlqvist Preservation for Modal Compact Hausdorff Spaces. In: Kennedy, J., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2017. Lecture Notes in Computer Science(), vol 10388. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55386-2_28

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  • DOI: https://doi.org/10.1007/978-3-662-55386-2_28

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