Overview
The morphological module used CFGs to compose morphemes into words. What happens when we use the same formalism to compose words into phrases and sentences? The change of empirical domain does not alter the definition of a CFG or the logic of CKY and Earley parsing.
It does change the pressure placed on the formalism. Syntactic grammars must represent long-distance dependencies, discontinuous constituents, and cross-serial patterns. Some of these can be encoded awkwardly in a CFG; others give rise to string languages that are not context-free (Shieber 1985). So we need to ask both whether a CFG generates the right strings and whether it assigns the right structures.
The formalisms proposed near that boundary do not define one universally agreed-upon class. Tree-Adjoining Grammars (TAGs), a restricted form of Combinatory Categorial Grammar (CCG), Head Grammars, and Linear Indexed Grammars are weakly equivalent (Vijay-Shanker and Weir 1994). Multiple Context-Free Grammars (MCFGs) and Linear Context-Free Rewriting Systems (LCFRSs) describe a larger family, while still permitting polynomial recognition for a fixed grammar (Seki et al. 1991). Formal versions of Minimalist Grammars can be translated into that MCFG/LCFRS family (Michaelis 2001).
We first apply CFGs to syntax and separate their weak generative capacity from the structures they assign. We then recast parsing as deduction, which supplies the control structure for the final project. With that machinery in place, we can ask where CFGs stop being adequate and how the mildly context-sensitive formalisms extend them.