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 Algol68 [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.

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.

Example

A well-known non-context-free language is

$\{a^n b^n a^n | n \ge 1\}.$

A two-level grammar for this language is the metagrammar

N ::= 1 | N1

X ::= a | b

together with grammar schema

Start ::= $\langle a^N \rangle\langle b^N \rangle\langle a^N \rangle$

$\langle X^{N1} \rangle$ ::= $\langle X^N \rangle X$

$\langle X^1 \rangle$ ::= X

See also

• Affix grammar
• Attribute grammar
• Van Wijngaarden grammar

External links

• Petersson, Kent (1990), "Syntax and Semantics of Programming Languages", Draft Lecture Notes, PDF text.