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Sequentialising Nested Systems

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Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2019)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11714))

Abstract

In this work, we investigate the proof theoretic connections between sequent and nested proof calculi. Specifically, we identify general conditions under which a nested calculus can be transformed into a sequent calculus by restructuring the nested sequent derivation (proof) and shedding extraneous information to obtain a derivation of the same formula in the sequent calculus. These results are formulated generally so that they apply to calculi for intuitionistic, normal modal logics and negative modalities.

Funded by the project Reasoning Tools for Deontic Logic and Applications to Indian Sacred Texts, CAPES, CNPq, START, FWF-ANR and TICAMORE (I 2982).

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Notes

  1. 1.

    Observe that, since in basic nested systems nested-like rules must have exactly one nesting in the premises or conclusion, only one hole is enough for describing both schemas and applications of rules. Compare with, e.g., the schematic nested rule for \(\mathsf {5}\) in [5].

  2. 2.

    Throughout, we will use n as a superscript, etc for indicating “nested”. Hence e.g., \(\rightarrow _R^n\) will be the designation of the implication right rule in the nesting framework.

  3. 3.

    We observe that, in [11] the basic sequent systems for \(\mathsf {K}\mathsf {B}\) and \(\mathsf {S5}\) were proved to be analytic (although not cut-free).

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Acknowledgments

We would like to thank Agata Ciabattoni for our fruitful discussions and the anonymous reviewers for their valuable comments.

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Authors and Affiliations

  1. Department of Mathematics, UFRN, Natal, Brazil

    Elaine Pimentel

  2. Institute of Logic and Computation, TU Wien, Vienna, Austria

    Revantha Ramanayake & Björn Lellmann

Authors
  1. Elaine Pimentel
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  2. Revantha Ramanayake
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  3. Björn Lellmann
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Corresponding author

Correspondence to Elaine Pimentel .

Editor information

Editors and Affiliations

  1. IBISC, Univ. Evry, Université Paris-Saclay, Evry, France

    Serenella Cerrito

  2. Middlesex University London, London, UK

    Andrei Popescu

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Pimentel, E., Ramanayake, R., Lellmann, B. (2019). Sequentialising Nested Systems. In: Cerrito, S., Popescu, A. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2019. Lecture Notes in Computer Science(), vol 11714. Springer, Cham. https://doi.org/10.1007/978-3-030-29026-9_9

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  • DOI: https://doi.org/10.1007/978-3-030-29026-9_9

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