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DOI: 10.4230/LIPIcs.FSCD.2018.10

Proof Nets for Bi-Intuitionistic Linear Logic

Authors: Gianluigi Bellin and Willem B. Heijltjes

Published in: LIPIcs, Volume 108, 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)

Abstract

Bi-Intuitionistic Linear Logic (BILL) is an extension of Intuitionistic Linear Logic with a par, dual to the tensor, and subtraction, dual to linear implication. It is the logic of categories with a monoidal closed and a monoidal co-closed structure that are related by linear distributivity, a strength of the tensor over the par. It conservatively extends Full Intuitionistic Linear Logic (FILL), which includes only the par. We give proof nets for the multiplicative, unit-free fragment MBILL-. Correctness is by local rewriting in the style of Danos contractibility, which yields sequentialization into a relational sequent calculus extending the existing one for FILL. We give a second, geometric correctness condition combining Danos-Regnier switching and Lamarche's Essential Net criterion, and demonstrate composition both inductively and as a one-off global operation.

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Gianluigi Bellin and Willem B. Heijltjes. Proof Nets for Bi-Intuitionistic Linear Logic. In 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 108, pp. 10:1-10:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bellin_et_al:LIPIcs.FSCD.2018.10,
  author =	{Bellin, Gianluigi and Heijltjes, Willem B.},
  title =	{{Proof Nets for Bi-Intuitionistic Linear Logic}},
  booktitle =	{3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)},
  pages =	{10:1--10:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-077-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{108},
  editor =	{Kirchner, H\'{e}l\`{e}ne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2018.10},
  URN =		{urn:nbn:de:0030-drops-91800},
  doi =		{10.4230/LIPIcs.FSCD.2018.10},
  annote =	{Keywords: proof nets, intuitionistic linear logic, contractibility, linear logic}
}

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Proof Nets for Bi-Intuitionistic Linear Logic

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