metalinear language

Recall that a linear grammar is a formal grammar G=(Σ,N,P,σ) whose productions are of the form Ax, where A is a terminal symbol, and x is a word over Σ, with at most one occurrence of a non-terminal symbol.

The conceptMathworldPlanetmath of a linear grammar can be generalized: define a k-linear grammar as a formal grammar G=(Σ,N,P,σ) such that every production in P has one of the three following forms:

  • Au,

  • AuBv,

  • σW,

where A,B are non-terminal symbols, u,v are terminal words, and W is a word over Σ with no more than k occurrences of non-terminal symbols, and none of which is the start symbol σ. Any k-linear grammar is context-free.

A languagePlanetmathPlanetmath is said to be k-linear if it can be generated by a k-linear grammar. Note that a language is 1-linear iff it is linear.

A language is said to be metalinear if it is k-linear for some positive integer k. In other words, if (k) denotes the family of k-linear languages, then the family () of metalinear langauges is


It is easy to see we have the following inclusions


where and are the families of regularPlanetmathPlanetmath and context-free languages respectively.

In fact, it can be shown that all of the inclusions above are strict, providing us with an infinite chain of families of languages between the regular languages and the context-free languages.


Title metalinear language
Canonical name MetalinearLanguage
Date of creation 2013-03-22 18:57:09
Last modified on 2013-03-22 18:57:09
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 68Q45
Classification msc 68Q70
Related topic LinearGrammar
Related topic LinearLanguage
Defines k-linear language