There are various criteria for the correctness of multiplicative proof nets. In Correctness of Multiplicative Proof Nets is Linear, Stefano Guerrini nicely describes few of them with an efficient algorithm.
We can erase the difference between par and tensor connectives and treat it as an interaction net with a single node (agent) type. In such a net we still can perform cut elimination faithfully as the procedure does not depend on the node labeling (by par and tensor). Is that right?
Assuming that's right let's consider the converse problem. Let's have an interaction net with single agent type that normalizes. Further assume that the normalized net has no vicious cycles.
Is it always possible to assign par-tensor labeling to the un-normalized net so that we obtain a correct proof net? Is there an efficient 'labeling' inference algorithm?