# 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.^{[1]} 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.

## Contents

## 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 , 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, 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 , 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

S

aSa

abSba

abbSbba

abbbba

and the corresponding GCFG production is

## Linear Context-free Rewriting Systems (LCFRSs)

Weir (1988)^{[1]} 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). LCFRS is a proper subclass of the GCFGs, i.e. it has strictly less computational power than the GCFGs as a whole.

On the other hand, LCFRSs are strictly more expressive than linear-indexed grammars and their weakly equivalent variant tree adjoining grammars (TAGs).^{[2]} Head grammar is another example of an LCFRS that is strictly less powerful than the class of LCFRSs as a whole.

LCFRS are weakly equivalent to (set-local) *multicomponent* TAGs (MCTAGs) and also with multiple context-free grammar (MCFGs [1]).^{[3]} and minimalist grammars (MGs). The languages generated by LCFRS (and their weakly equivalents) can be parsed in polynomial time.^{[4]}

## See also

## References

- ↑
^{1.0}^{1.1}Weir, David Jeremy (Sep 1988).*Characterizing mildly context-sensitive grammar formalisms*(PDF) (Ph.D.). Paper. AAI8908403. University of Pennsylvania Ann Arbor.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ Laura Kallmeyer (2010).
*Parsing Beyond Context-Free Grammars*. Springer Science & Business Media. p. 33. ISBN 978-3-642-14846-0.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ Laura Kallmeyer (2010).
*Parsing Beyond Context-Free Grammars*. Springer Science & Business Media. p. 35-36. ISBN 978-3-642-14846-0.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ Johan F.A.K. van Benthem; Alice ter Meulen (2010).
*Handbook of Logic and Language*(2nd ed.). Elsevier. p. 404. ISBN 978-0-444-53727-0.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>