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International Symposium on Logical Foundations of Computer Science

LFCS 2016: Logical Foundations of Computer Science pp 90–105Cite as

Multiple Conclusion Linear Logic: Cut Elimination and More

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9537))

Abstract

Full Intuitionistic Linear Logic (FILL) was first introduced by Hyland and de Paiva, and went against current beliefs that it was not possible to incorporate all of the linear connectives, e.g. tensor, par, and implication, into an intuitionistic linear logic. It was shown that their formalization of FILL did not enjoy cut-elimination by Bierman, but Bellin proposed a change to the definition of FILL in the hope to regain cut-elimination. In this note we adopt Bellin’s proposed change and give a direct proof of cut-elimination. Then we show that a categorical model of FILL in the basic dialectica category is also a LNL model of Benton and a full tensor model of Melliès’ and Tabareau’s tensorial logic. Lastly, we give a double-negation translation of linear logic into FILL that explicitly uses par in addition to tensor.

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Notes

  1. 1.

    The Agda development can be found at https://github.com/heades/cut-fill-agda.

  2. 2.

    This proof was formalized in the Agda proof assistant see the file https://github.com/heades/cut-fill-agda/blob/master/FullLinCat.agda.

  3. 3.

    We give a full proof in the formalization see the file https://github.com/heades/cut-fill-agda/blob/master/Tensorial.agda.

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Author information

Authors and Affiliations

  1. Computer and Information Sciences, Georgia Regents University, Augusta, GA, Georgia

    Harley Eades III

  2. Nuance Communications, Sunnyvale, CA, USA

    Valeria de Paiva

Authors
  1. Harley Eades III
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  2. Valeria de Paiva
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Correspondence to Harley Eades III .

Editor information

Editors and Affiliations

  1. City University of New York , New York, New York, USA

    Sergei Artemov

  2. Cornell University , Ithaca, New York, USA

    Anil Nerode

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Eades, H., de Paiva, V. (2016). Multiple Conclusion Linear Logic: Cut Elimination and More. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2016. Lecture Notes in Computer Science(), vol 9537. Springer, Cham. https://doi.org/10.1007/978-3-319-27683-0_7

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  • DOI: https://doi.org/10.1007/978-3-319-27683-0_7

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  • Online ISBN: 978-3-319-27683-0

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