It will be clear from this description that the ACG architecture is closely related to the compositional interpretation for categorial type logics as discussed in this artile. A key difference is the nature of the ‘abstract syntax’, i.e. the source calculus from which interpretations are homomorphically derived. In the case of the standard Lambek systems and the extended systems discussed in §3 above, the abstract syntax is a directional type logic; in the case of ACG, one finds LP and the linear lambda calculus both at the source and at the target end.  The debate as to whether structural properties of language have to be accounted for at the level of the abstract syntax has a long history, starting with Curry 1961; see Muskens 2007 for discussion.  The typelogical view accounts for word order universals at the level of its logical constants, i.e., the type-forming operations, and the laws that govern them.  The ACG view is more liberal in this respect: the derivation of surface form can be specified on a word-by-word basis. Whether this more relaxed connection between abstract syntax and surface realisation is desirable, is a matter of debate. An attractive feature of ACG that has not been investigated systematically within the typelogical setting is the connection between expressivity and order restrictions on the source constants and on the interpretive mapping.