Matrix grammar

A matrix grammar is a formal grammar in which instead of single productions, productions are grouped together into finite sequences. A production cannot be applied separately, it must be applied in sequence. In the application of such a sequence of productions, the rewriting is done in accordance to each production in sequence, the first one, second one etc. till the last production has been used for rewriting. The sequences are referred to as matrices.

Matrix grammar is an extension of context-free grammar, and one instance of a controlled grammar.

Formal definition

A matrix grammar is an ordered quadruple

where

The pairs are called productions, written as . The sequences are called matrices and can be written as

Let be the set of all productions appearing in the matrices of a matrix grammar . Then the matrix grammar is of type-, length-increasing, linear, -free, context-free or context-sensitive if and only if the grammar has the following property.

For a matrix grammar , a binary relation is defined; also represented as . For any , holds if and only if there exists an integer such that the words

over V exist and

If the above conditions are satisfied, it is also said that holds with as the specifications.

Let be the reflexive transitive closure of the relation . Then, the language generated by the matrix grammar is given by

Example

Consider the matrix grammar

where is a collection containing the following matrices:

These matrices, which contain only context-free rules generate the context-sensitive language

This example can be found on pages 8 and 9 of .

Properties

Let be the class of languages produced by matrix grammars, and MAT the class of languages produced by -free matrix grammars.

Open problems

It is not known whether there exist languages in which are not in MAT, and it is neither known whether contains languages which are not context-sensitive .

Footnotes

    References

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