Proof Nets for Bi-Intuitionistic Linear Logic

Authors Gianluigi Bellin, Willem B. Heijltjes

Thumbnail PDF


  • Filesize: 0.49 MB
  • 18 pages

Document Identifiers

Author Details

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

Cite AsGet BibTex

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)


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
  • proof nets
  • intuitionistic linear logic
  • contractibility
  • linear logic


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Gianluigi Bellin. Subnets of proof-nets in multiplicative linear logic with MIX. Mathematical Structures in Computer Science, 7(6):663-669, 1997. Google Scholar
  2. Gianluigi Bellin and Jacques van de Wiele. Subnets of proof-nets in MLL^-. In Advances in Linear Logic, pages 249-270, 1995. Google Scholar
  3. G.M. Bierman. A note on full intuitionistic linear logic. Annals of Pure and Applied Logic, 79(3):281-287, 1996. Google Scholar
  4. Torben Braüner and Valeria de Paiva. A formulation of linear logic based on dependency-relations. In 11th International Workshop on Computer Science Logic (CSL), 1997. Google Scholar
  5. Ranald Clouston, Jeremy Dawson, Rajeev Gore, and Alwen Tiu. Annotation-free sequent calculi for full intuitionistic linear logic. In LIPIcs-Leibniz International Proceedings in Informatics, volume 23. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2013. Google Scholar
  6. Robin Cockett and Robert Seely. Proof theory for full intuitionistic linear logic, bilinear logic, and MIX categories. Theory and Applications of Categories, 3(5):85-131, 1997. Google Scholar
  7. Robin Cockett and Robert Seely. Weakly distributive categories. Journal of Pure and Applied Algebra, 114:133-173, 1997. Google Scholar
  8. Vincent Danos. La Logique Linéaire appliquée à l'étude de divers processus de normalisation (principalement du Lambda-calcul). PhD thesis, Université Paris 7, 1990. Google Scholar
  9. Vincent Danos and Laurent Regnier. The structure of multiplicatives. Archive for Mathematical Logic, 28:181-203, 1989. Google Scholar
  10. Paulin Jacobé De Naurois and Virgile Mogbil. Correctness of linear logic proof structures is NL-complete. Theoretical Computer Science, 412(20):1941-1957, 2011. Google Scholar
  11. Harley Eades and Valeria de Paiva. Multiple conclusion linear logic: Cut elimination and more. In International Symposium on Logical Foundations of Computer Science (LFCS), pages 90-105, 2016. Google Scholar
  12. Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50(1):1-102, 1987. Google Scholar
  13. V. N. Grishin. On a generalization of the ajdukiewicz-lambek system. In Studies in Non-Classical Logics and Formal Systems, pages 315-343. Nauka, Moscow, 1983. Google Scholar
  14. Stefano Guerrini and Andrea Masini. Parsing MELL proof nets. Theoretical Computer Science, 254(1-2):317-335, 2001. Google Scholar
  15. Willem Heijltjes and Robin Houston. Proof equivalence in MLL is PSPACE-complete. Logical Methods in Computer Science, 12(1), 2016. Google Scholar
  16. Dominic J.D. Hughes. Simple free star-autonomous categories and full coherence. Journal of Pure and Applied Algebra, 216(11):2386-2410, 2012. Google Scholar
  17. Martin Hyland and Valeria de Paiva. Full intuitionistic linear logic. Annals of Pure and Applied Logic, 64(3):273-291, 1993. Google Scholar
  18. Yves Lafont. From proof nets to interaction nets. In J.-Y. Girard, Y. Lafont, and L. Regnier, editors, Advances in Linear Logic, volume 222 of London Mathematical Society Lecture Notes, pages 225-247. Cambridge University Press, 1995. Google Scholar
  19. François Lamarche. Proof nets for intuitionistic linear logic: Essential nets. Research report <inria-00347336>, INRIA, 2008. Google Scholar
  20. Joachim Lambek. Deductive systems and categories II. Lecture Notes in Mathematics, 86:76-122, 1969. Google Scholar
  21. Andrzej S. Murawski and C.-H. Luke Ong. Exhausting strategies, joker games and full completeness for IMLL with unit. Theoretical Computer Science, 294(1):269-305, 2003. Google Scholar
  22. Harold Schellinx. Some syntactical observations on linear logic. J. Logic and Computation, 1(4):537-559, 1991. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail