Two-level grammar

A two-level grammar is a formal grammar that is used to generate another formal grammar , such as one with an infinite rule set . This is how a Van Wijngaarden grammar was used to specify Algol 68 . 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.

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

See also

• Affix grammar
• Attribute grammar
• Van Wijngaarden grammar

References

1. Sintzoff, M. "Existence of van Wijngaarden syntax for every recursively enumerable set", Annales de la Société Scientifique de Bruxelles 2 (1967), 115-118.

External links

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