Abstract
We consider a class of graphs embedded in \(R^2\) as noncommutative proof-nets with an explicit exchange rule. We give two characterization of such proof-nets, one representing proof-nets as CW-complexes in a two-dimensional disc, the other extending a characterization by Asperti. As a corollary, we obtain that the test of correctness in the case of planar graphs is linear in the size of the data. Braided proof-nets are proof-nets for multiplicative linear logic with Mix embedded in \(R^3\). In order to prove the cut-elimination theorem, we consider proof-nets in \(R^2\) as projections of braided proof-nets under regular isotopy.
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