Regulated rewriting

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

Basic concepts

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

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

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

Basic concepts

Definition

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

Basic concepts

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

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.

References

  • Salomaa, Arto (1973) Formal languages. Academic Press, ACM monograph series
  • 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)
  • Dassow, Jürgen; Paun, G. 1990, Regulated Rewriting in Formal Language Theory ISBN 0387514147. Springer-Verlag New York, Inc. Secaucus, New Jersey, USA , Pages: 308. Medium: Hardcover.
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