Montague's Universal Grammar programme models compositionality as a structure-preserving mapping between a source and a target algebra. Abstract Categorial Grammar, introduced by de Groote in 2001, realizes this programme in the setting of MILL (multiplicative intuitionistic linear logic). In ACG, the derivations of an abstract source logic characterize the 'tectogrammatical' aspects of grammatical composition; the assembly of surface form and meaning is obtained as the image of tectogrammatical derivations under compositional translations.

ACG generates a hierarchy of grammars and the corresponding languages, G(n,m) and L(n,m), in terms of a double complexity measure: the maximal order of constants at the level of the source logic, and the maximal order of the image of source atoms under the compositional mappings. The Chomsky hierarchy and its MCS refinement is obtained from purely applicative source derivations, together with compositional mappings of increasing complexity.

Key notions to be covered:

• Multiplicative intuitionistic linear logic and the linear lambda calculus.
• Higher-order linear signatures and compositional mappings.
• The mildly context-sensitive hierarchy and the ACG G(n,m) hierarchy.
• Graphical calculus: linear logic proof nets and their underlying dynamic graph.

Sanne on the ACG encoding of TAG; Michiel on the ACG encoding of LCFG/MCFG.