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Proof Theory for Indexed Nested Sequents

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Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2017)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10501))

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Abstract

Fitting’s indexed nested sequents can be used to give deductive systems to modal logics which cannot be captured by pure nested sequents. In this paper we show how the standard cut-elimination procedure for nested sequents can be extended to indexed nested sequents, and we discuss how indexed nested sequents can be used for intuitionistic modal logics.

S. Marin—Supported by ERC Advanced Grant “ProofCert”.

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Notes

  1. 1.

    This is the variant of \(\mathsf {IK}\) first mentioned in [5] and [20] and studied in detail in [25]. There are many more variants of intuitionistic modal logic, e.g. [2, 6, 19, 22]. Another popular variant is constructive modal logic (e.g. [17]), which rejects axioms \(\mathsf {k_{3}}\)-\(\mathsf {k_{5}}\) in (7) and only allows \(\mathsf {k_{1}}\) and \(\mathsf {k_{2}}\). It has a different cut-elimination proof in nested sequents [1]. For this reason we work in this paper with \(\mathsf {IK}\) which allows all of \(\mathsf {k_{1}}\)–\(\mathsf {k_{5}}\).

  2. 2.

    Indeed, like \(\mathsf {iNK_2}\) and \(\mathsf {iNIK}\), Negri’s [18] system for classical logic \(\mathsf {K}\) can be seen as the classical variant of Simpson’s system [25] for intuitionistic logic \(\mathsf {IK}\). Then the same structural rules can be added to each system to extend it to geometric axioms, so in particular to Scott-Lemmon axioms.

  3. 3.

    One might consider this definition unsatisfactory as it is not a pure frame condition, but we have to leave a detailed study of this issue to future research.

  4. 4.

    We define the composition of two relations R, S on a set W as usual: \(R\circ S=\{(w,v)\mid \exists u.\;(wRu \wedge uSv)\} \). \(R^n\) stands for R composed n times with itself.

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  1. Inria, Saclay, France

    Sonia Marin & Lutz Straßburger

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  1. Sonia Marin
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  2. Lutz Straßburger
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Correspondence to Sonia Marin .

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  1. University of Manchester, Manchester, United Kingdom

    Renate A. Schmidt

  2. University of Brasília, Brasília D.F., Brazil

    Cláudia Nalon

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Marin, S., Straßburger, L. (2017). Proof Theory for Indexed Nested Sequents. In: Schmidt, R., Nalon, C. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2017. Lecture Notes in Computer Science(), vol 10501. Springer, Cham. https://doi.org/10.1007/978-3-319-66902-1_5

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  • DOI: https://doi.org/10.1007/978-3-319-66902-1_5

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