Proof Nets for First-Order Additive Linear Logic

Authors Willem B. Heijltjes, Dominic J. D. Hughes, Lutz Straßburger

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Author Details

Willem B. Heijltjes
  • University of Bath, United Kingdom
Dominic J. D. Hughes
  • Logic Group, UC Berkeley, USA
Lutz Straßburger
  • Inria Saclay, Palaiseau, France
  • LIX, École Polytechnique, Palaiseau, France


We would like to thank the anonymous referees for their constructive feedback. Dominic Hughes thanks his hosts, Wes Holliday and Dana Scott, at the UC Berkeley Logic Group.

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Willem B. Heijltjes, Dominic J. D. Hughes, and Lutz Straßburger. Proof Nets for First-Order Additive Linear Logic. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 22:1-22:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We present canonical proof nets for first-order additive linear logic, the fragment of linear logic with sum, product, and first-order universal and existential quantification. We present two versions of our proof nets. One, witness nets, retains explicit witnessing information to existential quantification. For the other, unification nets, this information is absent but can be reconstructed through unification. Unification nets embody a central contribution of the paper: first-order witness information can be left implicit, and reconstructed as needed. Witness nets are canonical for first-order additive sequent calculus. Unification nets in addition factor out any inessential choice for existential witnesses. Both notions of proof net are defined through coalescence, an additive counterpart to multiplicative contractibility, and for witness nets an additional geometric correctness criterion is provided. Both capture sequent calculus cut-elimination as a one-step global composition operation.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof theory
  • Theory of computation → Linear logic
  • linear logic
  • first-order logic
  • proof nets
  • Herbrand’s theorem


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