# Two-level grammar

A two-level grammar is a formal grammar that is used to generate another formal grammar [1], such as one with an infinite rule set [2]. This is how a Van Wijngaarden grammar was used to specify Algol 68 [3]. A context free grammar that defines the rules for a second grammar can yield an effectively infinite set of rules for the derived grammar. This makes such two-level grammars more powerful than a single layer of context free grammar, because generative two-level grammars have actually been shown to be Turing complete.[1]

Two-level grammar can also refer to a formal grammar for a two-level formal language, which is a formal language specified at two levels, for example, the levels of words and sentences.[citation needed]

## Example

A well-known non-context-free language is

${\displaystyle \{a^{n}b^{n}a^{n}|n\geq 1\}.}$

A two-level grammar for this language is the metagrammar

N ::= 1 | N1
X ::= a | b

together with grammar schema

Start ::= ${\displaystyle \langle a^{N}\rangle \langle b^{N}\rangle \langle a^{N}\rangle }$
${\displaystyle \langle X^{N1}\rangle }$  ::= ${\displaystyle \langle X^{N}\rangle X}$
${\displaystyle \langle X^{1}\rangle }$  ::= X