Generalized context-free grammar
Generalized Context-free Grammar (GCFG) is a grammar formalism that expands on context-free grammars by adding potentially non-context free composition functions to rewrite rules. Head grammar (and its weak equivalents) is an instance of such a GCFG which is known to be especially adept at handling a wide variety of non-CF properties of natural language.
A GCFG consists of two components: a set of composition functions that combine string tuples, and a set of rewrite rules. The composition functions all have the form , where is either a single string tuple, or some use of a (potentially different) composition function which reduces to a string tuple. Rewrite rules look like , where , , ... are string tuples or non-terminal symbols.
The rewrite semantics of GCFGs is fairly straight forward. An occurrence of a non-terminal symbol is rewritten using rewrite rules as in a context-free grammar, eventually yielding just compositions (composition functions applied to string tuples or other compositions). The composition functions are then applied, reducing successively reducing the tuples to a single tuple.
A simple translation of a context-free grammar into a GCFG can be performed in the following fashion. Given the grammar in (1), which generates the palindrome language , where is the string reverse of , we can define the composition function conc as in (2a) and the rewrite rules as in (2b).
The CF production of abbbba is
and the corresponding GCFG production is
 Linear Context-free Rewriting Systems (LCFRSs)
Weir (1988) describes two properties of composition functions, linearity and regularity. A function defined as is linear if and only if each variable appears at most once on either side of the =, making linear but not . A function defined as is regular if the left hand side and right hand side have exactly the same variables, making regular but not or .
A grammar in which all composition functions are both linear and regular is called a Linear Context-free Rewriting System (LCFRS), a subset of the GCFGs with strictly less computational power than the GCFGs as a whole, which is weakly equivalent to multicomponent Tree adjoining grammars. Head grammar is an example of an LCFRS that is strictly less powerful than the class of LCFRSs as a whole.
- Weir, David H. 1988. Characterizing mildly context-sensitive grammar formalisms. Dissertation, U Penn.