Secondary vector bundle structure

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.

Description

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 $f(⟨x_1,...,x_m⟩,⟨y_1,...,y_n⟩,...)=\gamma$, where $\gamma$ 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 $X\to f(Y,Z,...)$, where $Y$, $Z$, ... 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.

Example

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 $\{w{w}^R:w\in \{a,b{\}}^{*}\}$, where $w^R$ is the string reverse of $w$, we can define the composition function conc as in (2a) and the rewrite rules as in (2b).

1. $S\to ϵ\left|aSa\right|bSb$
2. 1. $conc(⟨x⟩,⟨y⟩,⟨z⟩)=⟨xyz⟩$
   2. $S\to conc(⟨ϵ⟩,⟨ϵ⟩,⟨ϵ⟩)|conc(⟨a⟩,S,⟨a⟩)|conc(⟨b⟩,S,⟨b⟩)$

The CF production of abbbba is

S

aSa

abSba

abbSbba

abbbba

and the corresponding GCFG production is

$S\to conc(⟨a⟩,S,⟨a⟩)$

$conc(⟨a⟩,conc(⟨b⟩,S,⟨b⟩),⟨a⟩)$

$conc(⟨a⟩,conc(⟨b⟩,conc(⟨b⟩,S,⟨b⟩),⟨b⟩),⟨a⟩)$

$conc(⟨a⟩,conc(⟨b⟩,conc(⟨b⟩,conc(⟨ϵ⟩,⟨ϵ⟩,⟨ϵ⟩),⟨b⟩),⟨b⟩),⟨a⟩)$

$conc(⟨a⟩,conc(⟨b⟩,conc(⟨b⟩,⟨ϵ⟩,⟨b⟩),⟨b⟩),⟨a⟩)$

$conc(⟨a⟩,conc(⟨b⟩,⟨bb⟩,⟨b⟩),⟨a⟩)$

$conc(⟨a⟩,⟨bbbb⟩,⟨a⟩)$

$⟨abbbba⟩$

Linear Context-free Rewriting Systems (LCFRSs)

Weir (1988) describes two properties of composition functions, linearity and regularity. A function defined as $f(x_1,...,x_n)=...$ is linear if and only if each variable appears at most once on either side of the =, making $f(x)=g(x,y)$ linear but not $f(x)=g(x,x)$. A function defined as $f(x_1,...,x_n)=...$ is regular if the left hand side and right hand side have exactly the same variables, making $f(x,y)=g(y,x)$ regular but not $f(x)=g(x,y)$ or $f(x,y)=g(x)$.

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.

References

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