## 6 Comparison with Certain Standard 2D Grammar Models  
A well-known class of two-dimensional picture array languages, called the family of context-sensitive matrix languages \((CSML)\), was introduced by [PERSON] et al. [50]. The corresponding class of grammars, known as \(CSMG\) involves two phases of generation with the first phase generating a string \(s\) over intermediate symbols and in the second phase all the intermediate symbols in \(s\) are rewritten in parallel by regular nonterminal rules of the form \(A\to a\), or together terminated by regular terminal rules of the form \(A\to a\), \((A,B\) are nonterminals and \(a\) is a terminal symbol), thus generating the columns of the picture array. The words in the first phase form a context-sensitive language. An extension, called \(TCSML\) of the family \(CSML\) was introduced in [51] by having in a \(CSMG\), an additional feature of tables of nonterminal rules or tables of terminal rules in the second phase. It was shown in [51] that \(CSML\subset TCSML\). We show here that there are picture array languages that cannot be in \(TCSML\) but can be generated by \(APRPS\) with two membranes. Theorem 6.1: \(AP_{2}(RPS)-TCSML\
eq\phi\)__  
Proof: Consider the picture array language \(L_{3}\) consisting of \((2m+1)\times(2m+1)\) arrays, \((m\geq 1)\), such that an array in \(L_{3}\) is of the form \((M_{1}\circ M_{2}\circ M_{3})\circ M_{4}\circ(M_{3}\circ M_{2}\circ M_{1})\) where \(M_{1}\), \(M_{3}\) are \(m\times m\) arrays over \(\{a\}\), \(\{b\}\) respectively, \(M_{2}\) is an \(m\times 1\) array over \(\{e\}\) and \(M_{4}\) is an \(1\times(2m+1)\) array over \(\{e\}\). A member of \(L_{3}\) is shown in Figure 4. The language \(L_{3}\) cannot belong to the family \(TCSML\). In fact in a grammar generating a picture array in a language in \(TCSML\), the number of rows above and below a distinct row of a generated array, cannot be maintained to be equal as the application of the tables of rules cannot be controlled. The language \(L_{3}\) is generated by the \(APRPS\)\(\Pi_{3}\) with two membranes where  
\[\Pi_{3}=(\{a,b,e\},[\,[\,[\,[\,[\,]_{2}\,]_{2}\,]_{1},F_{1},F_{2},R_{1},R_{2},2)\]  
with \(F_{1}=\{ \begin{array}{cc}a&b\\ b&a\end{array} \}\), \(F_{2}=\phi\). \(R_{1}\) consists of the column insertion tables \((c_{1},in)\), \((c_{2},in)\) and \(R_{2}\) consists of the row insertion tables \((r_{3},out)\), \((r_{4},here)\) where  
\[c_{1}=\{(a,ab,b),(b,ba,a)\},c_{2}=\{(a,e,b),(b,e,a)\},\]  
\[r_{3}=\{(a,^{t}ab,b),(b,^{t}ba,a)\},r_{4}=\{(a,e,b),(b,e,a),(e,e,e)\}.\]  
Starting with the axiom picture array in region 1, a computation in \(\Pi_{3}\) can start in this region with the applicable tables of rules. If the rules of column insertion table \(c_{1}\) are applied then \(ab\) will be inserted in every row between the two symbols \(a\) and \(b\) in two adjacent columns while \(ba\) will be inserted in every row between the two symbols \(b\) and \(a\) in the same two adjacent columns, thus adding two columns to the picture array. The array generated is sent to region 2. If in this region the rules of row insertion table \(r_{3}\) is applied then \(ab\) will be inserted in all columns between the symbols \(a\) and \(b\) in two adjacent rows while \(ba\) will be inserted between the symbols \(b\) and \(a\) in the same two adjacent rows, thus adding two rows to the picture array and the array is sent back to region 1. The process can  
Figure 3: A picture array generated by \(\Pi_{2}\). repeat. If at any time, the rules of the column insertion table \(c_{2}\) are applied in region 1, then a column of \(e^{\prime}s\) will be inserted between the two adjacent columns \({}^{t}a\cdots ab\cdots b\) and \({}^{t}b\cdots ba\cdots a\) and the array is sent to region 2. A correct sequence of application of the tables of rules is to apply now in region 2, the rules of the row table \(r_{4}\), thus generating a picture array of the form shown in Figure 4. Region 2 is the output region and arrays collected here form the picture array language generated which is \(L_{3}\).