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Proving Craig and Lyndon Interpolation Using Labelled Sequent Calculi

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Logics in Artificial Intelligence (JELIA 2016)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10021))

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Abstract

Interpolation is a fundamental logical property with applications in mathematics, computer science, and artificial intelligence. In this paper, we develop a general method of translating a semantic description of modal logics via Kripke models into a constructive proof of the Lyndon interpolation property (LIP) via labelled sequents. Using this method we demonstrate that all frame conditions representable as Horn formulas imply the LIP and that all 15 logics of the modal cube, as well as the infinite family of transitive Geach logics, enjoy the LIP.

This material is based upon work supported by the Austrian Science Fund (FWF) Lise Meitner Grant M 1770-N25.

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Notes

  1. 1.

    Unlike internal formalisms, external ones cannot generally be translated into formulas, typically because of the essential use of semantic elements, e.g., Kripke worlds.

  2. 2.

    The method also works for sequence- and set-based sequents.

  3. 3.

    NNF is used here to simplify the notation rather than out of necessity and means that negation is restricted to propositional atoms, creating two literals P and \(\overline{P}\) for each atom. Primary connectives are \(\wedge \), \(\vee \), \(\Box \), and \(\Diamond \). Negation \(\overline{A}\) is a function of a formula A defined via De Morgan laws. \(A \rightarrow B := \overline{A}\vee B\).

  4. 4.

    It also holds for \(l_i=0\): the empty conjunction is \(\top \) and \(\mathcal {M}, \left[ \!\left[ \mathsf {w}\right] \!\right] \Vdash \Diamond \top \).

  5. 5.

    For each eigenvariable \(\mathsf {y}_j\) we have collected all formulas labelled with \(\mathsf {y}_j\) within each disjunct into one labelled formula by transforming into if more than one formula has this label or by adding if no formula has.

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Acknowledgments

I am grateful to M. Fitting, whose idea started this interpolation project. I thank S. Negri for encouragement, V. Sikimić for procuring a source not available online, Y. Venema and M. Marx for valuable information on the non-constructive method. I am deeply indebted to B. Lellmann, who is always ready to listen and has provided many inspiring suggestions for improving this paper. I thank the anonymous reviewers for the suggestions on terminology.

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  1. Institut für Computersprachen, TU Wien, Vienna, Austria

    Roman Kuznets

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  1. Roman Kuznets
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Correspondence to Roman Kuznets .

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  1. Open University of Cyprus, Nicosia, Cyprus

    Loizos Michael

  2. University of Cyprus, Nicosia, Cyprus

    Antonis Kakas

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Kuznets, R. (2016). Proving Craig and Lyndon Interpolation Using Labelled Sequent Calculi. In: Michael, L., Kakas, A. (eds) Logics in Artificial Intelligence. JELIA 2016. Lecture Notes in Computer Science(), vol 10021. Springer, Cham. https://doi.org/10.1007/978-3-319-48758-8_21

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  • DOI: https://doi.org/10.1007/978-3-319-48758-8_21

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