Regulated rewriting

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 ${\displaystyle k(i)\geq 1}$, and ${\displaystyle 1\leq i\leq n}$, where each ${\displaystyle p_{i_{j}}1\leq j\leq k(i)}$, 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 ${\displaystyle L\rightarrow R}$. In these conditions the matrices can be written down as ${\displaystyle m_{i}=[L_{i_{1}}\rightarrow R_{i_{1}},...,L_{i_{k(i)}}\rightarrow R_{i_{k(i)}}]}$
4.- S is the start symbol

Definition
Let ${\displaystyle MG=(N,T,M,S)}$ 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 ${\displaystyle L(G)=\{a^{n}b^{n}c^{n}:n\geq 1\}}$ 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 ${\displaystyle M:}$ ${\displaystyle \left[S\rightarrow abc\right]}$, ${\displaystyle \left[S\rightarrow aAbBcC\right]}$, ${\displaystyle \left[A\rightarrow aA,B\rightarrow bB,C\rightarrow cC\right]}$, ${\displaystyle \left[A\rightarrow a,B\rightarrow b,C\rightarrow c\right]}$.

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