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:

Definition
A Matrix Grammar, ${\displaystyle MG}$, is a four-tuple ${\displaystyle G=(N,T,M,S)}$ where
1.- ${\displaystyle N}$ is an alphabet of non-terminal symbols
2.- ${\displaystyle T}$ is an alphabet of terminal symbols disjoint with ${\displaystyle N}$
3.- ${\displaystyle M=m_1,m_2,...,m_n}$ is a finite set of matrices, which are non-empty sequences ${\displaystyle m_i=[p_{i_1},...,p_{i_{k(i)}}]}$, with ${\displaystyle k(i)\ge 1}$, and ${\displaystyle 1\le i\le n}$, where each ${\displaystyle p_{i_j}1\le j\le k(i)}$, is an ordered pair ${\displaystyle p_{i_j}=(L,R)}$ being ${\displaystyle L\in (N\cup T{)}^{*}N(N\cup T{)}^{*},R\in (N\cup T{)}^{*}}$ these pairs are called "productions", and are denoted ${\displaystyle L\to R}$. In these conditions the matrices can be written down as ${\displaystyle m_i=[L_{i_1}\to R_{i_1},...,L_{i_{k(i)}}\to R_{i_{k(i)}}]}$
4.- S is the start symbol

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

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

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

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

Basic concepts

A Programmed Grammar is a pair ${\displaystyle (G,s)}$ where ${\displaystyle G=(N,T,P,S)}$ is a grammar and ${\displaystyle s,f:P\to 2^P}$ are the success and fail functions from the set of productions to the class of subsets of the set of productions.

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

Consider the CFG ${\displaystyle G=(N,T,P,S)}$ where ${\displaystyle N=\{S,A,B,C\}}$ is the non-terminal set, ${\displaystyle T=\{a,b,c\}}$ is the terminal set, and the productions set is defined as ${\displaystyle P=\{p_0,p_1,p_2,p_3,p_4,p_5,p_6\}}$ being ${\displaystyle p_0=S\to ABC}$ ${\displaystyle p_1=A\to aA}$, ${\displaystyle p_2=B\to bB}$, ${\displaystyle p_3=C\to cC}$ ${\displaystyle p_4=A\to a}$, ${\displaystyle p_5=B\to b}$, and ${\displaystyle p_6=C\to c}$. Clearly, ${\displaystyle L(G)=\{a^{*}{b}^{*}{c}^{*}\}}$. Now, considering the productions set ${\displaystyle P}$ as an alphabet (since it is a finite set), define the regular expression over ${\displaystyle P}$: ${\displaystyle e=p_0(p_1{p}_2{p}_3{)}^{*}(p_4{p}_5{p}_6)}$.

Combining the CFG grammar ${\displaystyle G}$ and the regular expression ${\displaystyle e}$, we obtain the CFGWRCL ${\displaystyle (G,e)=(G,p_0(p_1{p}_2{p}_3{)}^{*}(p_4{p}_5{p}_6))}$ which generates the language ${\displaystyle L(G)=\{a^n{b}^n{c}^n:n\ge 1\}}$.

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.

• 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.

• Dassow, Jürgen, Grammars with Regulated Rewriting. Lecture in the 5th PhD Program "Formal Languages and Applications", Tarragona, Spain, 2006.

• Abraham, S. 1965. Some questions of language theory, Proceedings of the 1965 International Conference On Computational Linguistics, pp. 1–11, Bonn, Germany,

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