Abstract Categorial Grammar.  The ACG framework (de Groote 2001a) is a meta-theory of compositional grammar architectures. ACGs are built on higher-order linear signatures Σ = (A,C,τ), where A and C are finite sets of atomic types and constants respectively, and τ a function assigning each constant a linear implicative type over A. Given a source signature Σ and a target signature Σ′, an interpretation is a mapping form Σ to Σ′ given by a pair of functions: η mapping the type atoms of Σ to linear implicative types of Σ′ and θ mapping the constants of Σ to well-typed linear lambda terms of Σ′ in a way that is compatible with the mapping on types. Using the terminology of compiler theory, one refers to the source and target signatures as the abstract vocabulary and the concrete vocabulary, respectively, and to the interpretive mapping as the lexicon. An ACG is then obtained by specifying an atomic type of Σ as the distinguished type of the grammar.