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Pomset logic and BV are both logics that extend multiplicative linear logic(with Mix) with a third connective that is self-dual and non-commutative.Whereas pomset logic originates from the study of coherence spaces and proofnets, BV originates from the study of series-parallel orders, cographs, andproof systems. Both logics enjoy a cut-admissibility result, but for neitherlogic can this be done in the sequent calculus. Provability in pomset logic canbe checked via a proof net correctness criterion and in BV via a deep inferenceproof system. It has long been conjectured that these two logics are the same.  In this paper we show that this conjecture is false. We also investigate thecomplexity of the two logics, exhibiting a huge gap between the two. Whereasprovability in BV is NP-complete, provability in pomset logic is$\Sigma_2^p$-complete. We also make some observations with respect to possiblesequent systems for the two logics.

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