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A Focus System for the Alternation-Free \(\mu \)-Calculus

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Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2021)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12842))

Abstract

We introduce a cut-free sequent calculus for the alternation-free fragment of the modal \(\mu \)-calculus. This system allows both for infinite and for finite, circular proofs and uses a simple focus mechanism to control the unravelling of fixpoints along infinite branches. We show that the proof system is sound and complete for the set of guarded valid formulas of the alternation-free \(\mu \)-calculus.

The authors want to thank the anonymous reviewers for many helpful comments.

J. Marti—The research of this author has been made possible by a grant from the Dutch Research Council NWO, project nr. 617.001.857.

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Notes

  1. 1.

    The rule \(\mathsf {U}\) is not really needed—in fact we prove completeness without it. We include \(\mathsf {U} \) because of its convenience for constructing proofs.

  2. 2.

    Note that since we assume guardedness, the principal formula is different from its residuals.

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Authors and Affiliations

  1. ILLC, University of Amsterdam, P.O. Box 94242, 1090 GE, Amsterdam, The Netherlands

    Johannes Marti & Yde Venema

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  1. Johannes Marti
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  2. Yde Venema
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Correspondence to Yde Venema .

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Editors and Affiliations

  1. University of Birmingham, Birmingham, UK

    Anupam Das

  2. University of Genoa, Genoa, Italy

    Sara Negri

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Marti, J., Venema, Y. (2021). A Focus System for the Alternation-Free \(\mu \)-Calculus. In: Das, A., Negri, S. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2021. Lecture Notes in Computer Science(), vol 12842. Springer, Cham. https://doi.org/10.1007/978-3-030-86059-2_22

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  • DOI: https://doi.org/10.1007/978-3-030-86059-2_22

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