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:

Contents

[edit] Matrix Grammars

[edit] Basic concepts

Definition
A Matrix Grammar, MG, is a four-tuple G = (N,T,M,S) where
1.- N is an alphabet of terminal symbols
2.- T is an alphabet of terminal symbols disjoint with T
3.- M = m1,m2,...,mn is a finite set of matrices, which are non-empty sequences m_{i} = [p_{i},...,p_{i_{k(i)}}], with k(i)\geq 1, and 1 \leq i \leq n, where each p_{i_{j}} 1\leq j\leq k(i), is an ordered pair p_{i_{j}} = (L, R) being L \in (N \cup T)^*N(N\cup T)^*, R \in (N\cup T)^* these pairs are called "productions", and are denoted L\rightarrow R. In these conditions the matrices can be written down as m_i = [L_{i_{1}}\rightarrow R_{i_{1}},...,L_{i_{k(i)}}\rightarrow R_{i_{k(i)}}]
4.- S is the start symbol

Definition
Let MG = (N,T,M,S) be a matrix grammar and let P the collection of all productions on matrices of MG. We said that MG is of type i according to Chomsky's hierarchy with i = 0,1,2,3, or "increasing length" or "linear" or "without λ-productions" if and only if the grammar G = (N,T,P,S) has the corresponding property.

[edit] The classical example (taked from [5] with change of nonterminals names)

The context-sensitive language L(G) = \{ a^nb^nc^n : n\geq 1\} is generated by the CFMG G = (N,T,M,S) where N = {S,A,B,C} is the non-terminal set, T = {a,b,c} is the terminal set, and the set of matrices is defined as M: \left[S\rightarrow abc\right], \left[S\rightarrow aAbBcC\right], \left[A\rightarrow aA,B\rightarrow bB,C\rightarrow cC\right], \left[A\rightarrow a,B\rightarrow b,C\rightarrow c\right].

[edit] Time Variant Grammars

Basic concepts
Definition
A Time Variant Grammar is a pair (G,v) where G = (N,T,P,S) is a grammar and v: \mathbb{N}\rightarrow 2^{P} is a function from the set of natural numbers to the class of subsets of the set the productions.

[edit] Programmed Grammars

Basic concepts

[edit] Definition

A Programmed Grammar is a pair (G,s) where G = (N,T,P,S) is a grammar and s: P\rightarrow 2^{P} is a function from the set of productions to the class of subsets of the set the productions.

[edit] Grammars with regular control language

[edit] Basic concepts

Definition
A Grammar With Regular Control Language, GWRCL, is a pair (G,e) where G = (N,T,P,S) is a grammar and e is a regular expression over the alphabet of the set the productions.

[edit] A naive example

Consider the CFG G = (N,T,P,S) where N = {S,A,B,C} is the non-terminal set, T = {a,b,c} is the terminal set, and the productions set is defined as P = {p0,p1,p2,p3,p4,p5,p6} being p_0 = S\rightarrow ABC p_1 = A\rightarrow aA, p_2 = B\rightarrow bB, p_3 = C\rightarrow cC p_4 = A\rightarrow a, p_5 = B\rightarrow b, and p_6 = C\rightarrow c. Clearly, L(G) = {a * b * c * }. Now, considering the productions set P as an alphabet (since it is a finite set), define the regular expression over P: e = p0(p1p2p3) * (p4p5p6).

Combining the CFG grammar G and the regular expression e, we obtain the CFGWRCL (G,e) = (G,p0(p1p2p3) * (p4p5p6)) which generates the language L(G) = \{ a^nb^nc^n : n\geq 1\}.


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.

[edit] Sources

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

[2] G. Rozenberg, A. Salomaa, (eds.) Handbook of formal languages Berlin; New York : Springer, 1997 ISBN 3540614869 (set) (3540604200 : v. 1; 3540606483 : v. 2; 3540606491: v. 3)

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

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

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