Completeness of LG with respect to this interpretation is proved in Kurtonina and Moortgat 2010.  The minimal symmetric system (without interaction principles) has fission R⊕ and merge R⊗ as distinct relations, without imposing restrictions on their interpretation.  In the presense of the interaction principles, their interpretation is related in terms of frame constraints. The distributivity principle (A ⦸ B) ⊗ C ⊢ A ⦸ (B ⊗ C), for example, corresponds to the constraint that for every x,y,z,w,v, if we have a configuration R⊗xyz and R⊕vwy, then there exists an alternative internal point t such that R⊕twx and R⊗tvz.  For decidability, Moortgat (2009) gives a sequent presentation of LG in the format of a Display Calculus allowing cut-free proof search.