Proof Nets, Coends and the Yoneda Isomorphism

TL;DR

This paper introduces a compact proof net representation for a fragment of Second Order Multiplicative Linear Logic, using the Yoneda isomorphism and a new equivalence relation called re-witnessing, to characterize proof equivalence via coends.

Contribution

It develops a novel proof net representation for MLL2 related to Yoneda isomorphism and defines re-witnessing to characterize proof equivalence generated by coends.

Findings

Re-witnessing characterizes proof equivalence in the fragment.

A compact proof net representation for MLL2 is provided.

The approach adapts the rewiring method from coherence results.

Abstract

Proof nets provide permutation-independent representations of proofs and are used to investigate coherence problems for monoidal categories. We investigate a coherence problem concerning Second Order Multiplicative Linear Logic (MLL2), that is, the one of characterizing the equivalence over proofs generated by the interpretation of quantifiers by means of ends and coends. We provide a compact representation of proof nets for a fragment of MLL2 related to the Yoneda isomorphism. By adapting the "rewiring approach" used in coherence results for star-autonomous categories, we define an equivalence relation over proof nets called "re-witnessing". We prove that this relation characterizes, in this fragment, the equivalence generated by coends.