## 2 Preliminaries
### Picture-Insertion System  
A picture-insertion system has been considered in [41] with insertion in a picture array being done between columns or between rows, extending the notion of insertion in words [40]. We recall the definition of this system [41] with a special case, namely, with \"contexts\" of length one in the rules and call this system as a restricted picture-insertion system. **Definition 1**. : _A restricted picture-insertion system \((RPIS)\) is a 4-tuple \(\Gamma=(\Sigma,I_{c},I_{r},A)\) where_  
1. \(\Sigma\) _is an alphabet;_
2. \(I_{c}=\{c_{i}\ |\ 1\leq i\leq m\},\,(m\geq 1)\) _where_ \(c_{i}\)_, called a column insertion table, is a finite set of column insertion rules with alphabetic contexts of the form_ \((a,\alpha,b)\)_,_ \(a,b\in\Sigma\cup\{\lambda\},\alpha\in\Sigma^{*}\) _such that for any two rules_ \((a_{1},\alpha,b_{1}),(a_{2},\beta,b_{2})\) _in_ \(c_{i}\)_, we have_ \(|\alpha|=|\beta|\) _and either both the left contexts_ \(a_{1}\) _and_ \(a_{2}\) _are in_ \(\Sigma\) _(likewise the right contexts_ \(b_{1}\) _and_ \(b_{2}\) _are in_ \(\Sigma\)_) or both are_ \(\lambda\) _;_
3. \(I_{r}=\{r_{j}\ |\ 1\leq j\leq n\},\,(n\geq 1)\) _where_ \(r_{j}\)_, called a row insertion table, is a finite set of row insertion rules with alphabetic contexts of the form_ \((d,{}^{t}\gamma,e)\)_,_ \(d,e\in\Sigma\cup\{\lambda\},\gamma\in\Sigma^{*}\) _such that for any two rules_ \((d_{1},{}^{t}\gamma,e_{1}),(d_{2},{}^{t}\delta,e_{2})\) _in_ \(r_{j}\)_, we have_ \(|\gamma|=|\delta|\) _and either both the up contexts_ \(d_{1}\) _and_ \(d_{2}\) _are in_ \(\Sigma\) _(likewise the down contexts_ \(e_{1}\) _and_ \(e_{2}\) _are in_ \(\Sigma\)_) or both are_ \(\lambda\)_;_
4. \(A\subseteq\Sigma^{**}-\{\lambda\}\) _is a finite set of axiom arrays._  
For picture arrays \(P,Q\in\Sigma^{**}\), a one-step derivation in the \(RPIS\)\(\Gamma\), denoted by \(P\Rightarrow Q\), yields \(Q\) from \(P\) whenever either \((i)\) or \((ii)\) holds: \((i)\) if \({}^{t}a_{1}\cdots a_{m}\) and \({}^{t}b_{1}\cdots b_{m}\) are two adjacent columns, namely, columns \(j\) and \(j+1\) for some \(j,1\leq j\leq n-1\), in the picture array \(P\) of size \(m\times n\), and if \((a_{i},a_{i},b_{i}),1\leq i\leq m\) are column insertion rules in a column insertion table in \(I_{c}\), then the rules can be applied in parallel by inserting \(a_{i}\) in the \(i\)th row between \(a_{i}\) and \(b_{i}\) for all \(i\), \(1\leq i\leq m\) or \((ii)\) if \(d_{1}\cdots d_{n}\) and \(e_{1}\cdots e_{n}\) are two adjacent rows, namely, rows \(k\) and \(k+1\) for some \(k,1\leq k\leq m-1\), in the picture array \(P\) of size \(m\times n\), and if \((d_{i},{}^{t}\beta_{i},e_{i}),1\leq i\leq n\) are row insertion rules in a row insertion table in \(I_{r}\), then the rules can be applied in parallel by inserting \({}^{t}\beta_{i}\) in the \(i\)th column between \(e_{i}\) and \(d_{i}\) for \(1\leq i\leq n\). Likewise insertions to the immediate left or right (respy: immediate up or down) of a column (respy. row) in the picture array \(P\) can be defined by requiring the corresponding left or right or up or down contexts in the rules used, to be \(\lambda\). The picture language \(L(\Gamma)\) generated by \(\Gamma\) consists of picture arrays derived in one or more finite number of derivation steps starting with an axiom array in \(A\). The family of picture languages generated by \(RPIS\) is denoted by \(RPIL\). **Example 1**. : _Consider the \(RPIS\)\(\Gamma_{1}=(\Sigma,I_{c},I_{r},A)\) where \(\Sigma=\{a,b\},\,I_{c}=\{c_{1}\},I_{r}=\{r_{1}\},\) where_  
\[c_{1}=\{(a,ab,b)\},r_{1}=\{(a,a,\lambda),(b,b,\lambda)\}\]  
_and \(A\) consists of the array \( \begin{array}{cc}a&b\\ a&b\end{array} .\)_  
\(\Gamma_{1}\) generates a picture language \(L_{1}\) consisting of picture arrays \(X\circ Y\) with \(X\) over \(a\) and \(Y\) over \(b\) and \(|X|_{r}=|Y|_{r},|X|_{c}=|Y|_{c}\). A member of \(L_{1}\) is shown in Figure 1.  
**Example 2**.