With respect to computational complexity, Moot (2008) establishes a correspondence between Lexicalized Tree Adjoining Grammars on the one hand, and categorial grammars with the multimodal extraction postulates of §3.1 and a restricted set of LG grammars on the other. For these grammars he obtains a polynomial parsability result via a translation into Hyperedge Replacement Grammars.  In the case of LG, the restriction requires the coimplications ⊘,⦸ to occur in matching pairs in lexical type assignments.  The lexicon of the generalized MIX construction of Melissen 2009, and the type assignment used for quantifier phrases in the analysis of scope construal in Bernardi and Moortgat 2010, do not respect this restriction. For the general case of LG with the interaction principles of §3.3, Bransen (2010) establishes NP-completeness.  The position of the discontinuous calculi of §3.2 in this hierarchy has to be determined: they recognize more than the context-free languages, but it is not clear whether they stay within the mildly context-sensitive family.