Regulated rewriting

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Regulated rewriting is a specific area of formal languages studying grammatical systems which are able to take some kind of control over the production applied in a derivation step. For this reason, the grammatical systems studied in Regulated Rewriting theory are also called "Grammars with Controlled Derivations". Among such grammars can be noticed:

Matrix Grammars[edit]

Basic concepts[edit]

Definition
A Matrix Grammar, , is a four-tuple where
1.- is an alphabet of non-terminal symbols
2.- is an alphabet of terminal symbols disjoint with
3.- is a finite set of matrices, which are non-empty sequences , with , and , where each , is an ordered pair being these pairs are called "productions", and are denoted . In these conditions the matrices can be written down as
4.- S is the start symbol

Definition
Let be a matrix grammar and let the collection of all productions on matrices of . We said that is of type i according to Chomsky's hierarchy with , or "increasing length" or "linear" or "without -productions" if and only if the grammar has the corresponding property.

The classic example[edit]

Note: taken from Abraham 1965, with change of nonterminals names

The context-sensitive language is generated by the where is the non-terminal set, is the terminal set, and the set of matrices is defined as , , , .

Time Variant Grammars[edit]

Basic concepts
Definition
A Time Variant Grammar is a pair where is a grammar and is a function from the set of natural numbers to the class of subsets of the set of productions.

Programmed Grammars[edit]

Basic concepts

Definition[edit]

A Programmed Grammar is a pair where is a grammar and are the success and fail functions from the set of productions to the class of subsets of the set of productions.

Grammars with regular control language[edit]

Basic concepts[edit]

Definition
A Grammar With Regular Control Language, , is a pair where is a grammar and is a regular expression over the alphabet of the set of productions.

A naive example[edit]

Consider the CFG where is the non-terminal set, is the terminal set, and the productions set is defined as being , , , , and . Clearly, . Now, considering the productions set as an alphabet (since it is a finite set), define the regular expression over : .

Combining the CFG grammar and the regular expression , we obtain the CFGWRCL which generates the language .

Besides there are other grammars with regulated rewriting, the four cited above are good examples of how to extend context-free grammars with some kind of control mechanism to obtain a Turing machine powerful grammatical device.

Sources[edit]

[1] Salomaa, Arto (1973) Formal languages Academic Press, ACM monograph series

[2] Rozenberg, G.; Salomaa, A. (eds.) 1997 Handbook of formal languages Berlin; New York : Springer ISBN 3-540-61486-9 (set) (3540604200 : v. 1; 3540606483 : v. 2; 3540606491: v. 3)

[3] Dassow, Jurgen; Paun, G. 1990 Regulated Rewriting in Formal Language Theory ISBN 0387514147. Springer-Verlag New York, Inc. Secaucus, NJ, USA Pages: 308. Medium: Hardcover.

[4] Dassow, Jurgen; von-Guericke, Otto Grammars with Regulated Rewriting Available at: [1] and [2] ([3])

[5] Abraham, S. 1965. "Some questions of language theory", Proceedings of the 1965 International Conference On Computational Linguistics, pp 1 – 11, Bonn, Germany Available at: [4]