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 proof nets, BV originates from the study of series-parallel orders, cographs, and proof systems. Both logics enjoy a cut-admissibility result, but for neither logic can this be done in the sequent calculus. Provability in pomset logic can be checked via a proof net correctness criterion and in BV via a deep inference proof 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 the complexity of the two logics, exhibiting a huge gap between the two. Whereas provability in BV is NP-complete, provability in pomset logic is $\Sigma_2^p$-complete. We also make some observations with respect to possible sequent systems for the two logics.

翻译：Pomset逻辑和BV都是通过引入一个自对偶且非交换的第三连接词来扩展乘法线性逻辑（含Mix）的逻辑体系。Pomset逻辑源于相干空间和证明网的研究，而BV则源于串并联序、余图及证明系统的研究。这两种逻辑均具有切割可容许性，但都无法在相继式演算中实现这一性质。Pomset逻辑的可证明性可通过证明网正确性准则检验，BV的可证明性则需通过深层推理证明系统验证。长久以来存在一个猜想认为这两种逻辑是等价的。本文证明该猜想不成立。我们进一步探究了两种逻辑的复杂性，揭示出二者间存在巨大差距：BV的可证明性为NP完全问题，而Pomset逻辑的可证明性为$\Sigma_2^p$完全问题。此外，本文还对两种逻辑可能对应的相继式系统进行了若干讨论。