Unique perfect matchings, forbidden transitions and proof nets for linear logic with Mix

A System of Interaction and Structure III: The Complexity of BV and Pomset Logic

“…The existence or non-existence of ae-cycles in RB-dicographs will play a central role in this paper. We note in passing that in the classical theory of matchings in undirected graphs, the absence of ae-cycles admits alternative characterizations and entails deep structural properties, which have been applied to MLL+mix proof nets [Ret03,Ngu20].…”

“…We are now going to fill the steps outlined above to prove the NP-hardness part of Theorem 4.13. The reduction step extends the "proofification" construction from [Ngu20] that sends undirected perfect matchings to MLL+mix pre-proof nets. For the sake of clarity, we present our new reduction, with the same name, as a map from arbitrary RB-digraphs to balanced pomset logic sequents.…”

“…This is still a correct net (replacing ◁ by removes an edge, so it can destroy ae-cycles, not create them), in fact G ′ is an MLL+mix proof net. There exists an MLL+mix sequent proof whose inductive translation (desequentialization) is G ′ -this is the sequentialization theorem (a result involving non-trivial combinatorics, originating in [Gir87]; see [Ret03,Ngu20] for a discussion of its "equivalence" with earlier results in mainstream graph theory). Next, in this sequentialization, replace the relevant occurrences of by ◁; we obtain a Slavnov pre-proof (since in this system the rule for ◁ is the same as the MLL+mix rule for ) which lifts uniquely to a decorated pre-proof.…”