In the ACG setting, one can model the syntax-semantics interface in terms of the abstract versus object vocabulary distinction. But one can also obtain surface form as the result of an interpretive mapping, using the canonical λ term encodings of strings and trees and operations on them. ACG has given rise to an interesting complexity hierarchy for rewriting grammar formalisms encoded as ACGs: context-free grammars, tree-adjoining grammars, etc.; see for example de Groote and Pogodalla 2004.  Expressive power of these formalisms is measured in terms of the maximal order of the constants in the abstract vocabulary and of the object types interpreting the atomic abstract types. The study of ACG encodings of typelogical systems proper has started with Retoré and Salvati 2010; these authors present an ACG construction for (product-free) NL.