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Proof Nets for Bi-Intuitionistic Linear Logic

Authors Gianluigi Bellin, Willem B. Heijltjes

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Gianluigi Bellin
  • Università di Verona, Verona, Italy
Willem B. Heijltjes
  • University of Bath, Bath, United Kingdom

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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)
https://doi.org/10.4230/LIPIcs.FSCD.2018.10

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Linear logic
  • Theory of computation → Proof theory
Keywords
  • proof nets
  • intuitionistic linear logic
  • contractibility
  • linear logic

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