On Some Closure Properties of nc-eNCE Graph Grammars

Consider two $nc-eNCE$ graph grammars $ncG_{1}=(\Sigma_{1},\Delta_{1},\Gamma_{1},\Omega_{1},P_{1},S_{1},$ $R_{1}(P_{1}))$ and $ncG_{2}=(\Sigma_{2},\Delta_{2},\Gamma_{2},\Omega_{2},P_{2},S_{2},R_{2}(P_{2}))$ generating graph languages $L_{1}$ and $L_{2}$ respectively. Assuming that $S$ is a new symbol that does not appear in $\Sigma_{1}\cup\Sigma_{2}$, we define a new $nc-eNCE$ graph grammar $ncG$ capable of generating the language $L=L_{1}UL_{2}$ as follows.

,

• $\Sigma=\Sigma_{1}\cup\Sigma_{2}\cup\{S\}$

• $\Delta=\Delta_{1}\cup\Delta_{2}$

• $\Gamma=\Gamma_{1}\cup\Gamma_{2}$

• $\Omega=\Omega_{1}\cup\Omega_{2}$

• $P=P_{1}\cup P_{2}\cup\{p_{01},p_{02}\}$

• R(P)= $[p_{01}.R_{1}(P_{1})]+[p_{02}.R_{2}(P_{2})]$

Here $S_{1}$ and $S_{2}$ are the start graphs of $ncG_{1}$ and $ncG_{2}$ respectively. Clearly any graph in $L_{1}$ can be generated using $[p_{01}.R_{1}(P_{1})]$ while any graph in $L_{2}$ can be generated using $[p_{02}.R_{2}(P_{2})]$. ∎

Consider two $nc-eNCE$ graph grammars $ncG_{1}=(\Sigma_{1},\Delta_{1},\Gamma_{1},\Omega_{1},P_{1},S_{1},$ $R_{1}(P_{1}))$ and $ncG_{2}=(\Sigma_{2},\Delta_{2},\Gamma_{2},\Omega_{2},P_{2},S_{2},R_{2}(P_{2}))$ generating graph languages $L_{1}$ and $L_{2}$ respectively. Without loss of generality we can assume that $P_{1}$ and $P_{2}$ have no elements in common.We define a new $nc-eNCE$ graph grammar $ncG$ capable of generating the language $L=L_{1}\oplus L_{2}$ as follows.

,

• $\Sigma=\Sigma_{1}\cup\Sigma_{2}\cup\{S\}$

• $\Delta=\Delta_{1}\cup\Delta_{2}$

• $\Gamma=\Gamma_{1}\cup\Gamma_{2}$

• $\Omega=\Omega_{1}\cup\Omega_{2}$

• $P=P_{1}\cup P_{2}\cup\{p_{01}\}$

• $R(P)=p_{01}.R_{1}(P_{1}).R_{2}(P_{2})$

The start graphs $S_{1}$ and $S_{2}$ of $ncG_{1}$ and $ncG_{2}$ are initially generated using $p_{01}$. This is followed by $R_{1}(P_{1})$ generating $G_{1}\in L_{1}$ and $R_{2}(P_{2})$ generating $G_{2}\in L_{2}$. Hence it is clear that $R(P)$ generates $L_{1}\oplus L_{2}$. ∎

Consider an $nc-eNCE$ graph grammar $ncG_{1}=(\Sigma_{1},\Delta_{1},\Gamma_{1},\Omega_{1},P_{1},S_{1},$ $R_{1}(P_{1}))$ generating the graph language $L_{1}$. We define a new $nc-eNCE$ graph grammar $ncG$ capable of generating the language $L=L_{1}^{\oplus}$ as follows.

,

• $\Sigma=\Sigma_{1}\cup\{S\}$

• $\Delta=\Delta_{1}$

• $\Gamma=\Gamma_{1}$

• $\Omega=\Omega_{1}$

• $P=P_{1}\cup\{p_{01},p_{02}\}$

• $R(P)=(p_{01}.R_{1}(P_{1}))^{*}.p_{02}$

Initially the production $p_{01}$ replaces the start node $S$ of $ncG$ with a copy of the start graph $S_{1}$ of $ncG_{1}$ and the node $S$ itself. Now $R_{1}(P_{1})$ replaces $S_{1}$ by a graph $G_{1}\in L_{1}$. The rules $p_{01}$ and $R_{1}(P_{1})$ are recursively applied to further generate new components of the resultant graph. Finally the derivation terminates when $p_{02}$ is applied. Hence it is clear that $R(P)$ generates $L_{1}^{\oplus}$. ∎

Consider two $nc-eNCE$ graph grammars $ncG_{1}=(\Sigma_{1},\Delta_{1},\Gamma_{1},\Omega_{1},P_{1},S_{1},$ $R_{1}(P_{1}))$ and $ncG_{2}=(\Sigma_{2},\Delta_{2},\Gamma_{2},\Omega_{2},P_{2},S_{2},R_{2}(P_{2}))$ generating graph languages $L_{1}$ and $L_{2}$ respectively. Without loss of generality we can assume that $P_{1}$ and $P_{2}$ have no elements in common. We begin our construction of the required grammar by carrying out the following steps.

1. 1.

   Replace each $p_{j}^{*}$ in the regular expressions $R_{i}(P_{i})$ by the equivalent expression $(p_{j}^{*}p_{j}+\varepsilon)$ and simplifying so that $R_{i}(P_{i})$ is now of the form $R_{i1}(P_{i})+R_{i2}(P_{i})+\cdots R_{in_{i}}(P_{i})$. This ensures that each $R_{ik}(P_{i})$ does not contain a the operator ’$+$’.

2. 2.

   Initialize ${P^{\prime}}_{1}=P_{1}$. For each $R_{1k}(P_{1}),1\leq k\leq n_{1}$, do the following

   1. (a)

      Since every graph generated by $R_{1k}(P_{1})$ is non null it contains atleast one vertex with a terminal label. Identify the last production rule $p_{1l}$ in the series $R_{1k}(P_{1})$ which has atleast one node labelled with a terminal symbol, say ’$a$’ on the right hand side.

   2. (b)

      Create a copy of $p_{1l}$ in which one occurrence of ’$a$’ is replaced by a new terminal symbol ’$x$’, $x\notin(\Delta_{1}\cup\Delta_{2})$ and label this production ${p^{\prime}}_{1l}$. The connection instructions corresponding to ${p^{\prime}}_{1l}$ includes all the instructions in ${p}_{1l}$. In addition every connection instruction which contains ’$a$’ is duplicated and ’$x$’ is inserted instead of ’$a$’. Update ${P^{\prime}}_{1}$ as ${P^{\prime}}_{1}={P^{\prime}}_{1}\cup{p^{\prime}}_{1l}$.

3. 3.

   Repeat Step 2 for $R_{2k}(P_{2}),1\leq k\leq n_{2}$ with the terminal symbol ’$b$’ on the right hand side of $p_{1l}$ replaced by $y$, $y\notin(\Delta_{1}\cup\Delta_{2})$ label this production ${p^{\prime}}_{1l}$. The connection instructions corresponding to ${p^{\prime}}_{1l}$ includes all the instructions in ${p}_{1l}$. In addition every connection instruction which contains ’$b$’ is duplicated and ’$y$’ is inserted instead of ’$b$’.One more jumping connection instruction of the form $(y,\alpha,x)$ is added to this set. Update ${P^{\prime}}_{2}$ as ${P^{\prime}}_{2}={P^{\prime}}_{2}\cup{p^{\prime}}_{1l}$ .

4. 4.

   Replace the last occurrence of $p_{il}$ with ${p^{\prime}}_{il}$ in each $R_{ik}(P_{i})$ to obtain ${R^{\prime}}_{ik}(P_{i})$. The new regular control is given by

   $${R^{\prime}}_{i}({P^{\prime}}_{i})={R^{\prime}}_{i1}({P^{\prime}}_{i})+{R^{\prime}}_{i2}({P^{\prime}}_{i})+\cdots{R^{\prime}}_{in_{i}}({P^{\prime}}_{i})$$

.

We now define a new $nc-eNCE$ graph grammar $ncG$ capable of generating the language $L$ such that $U(L)=U(L_{1}\odot L_{2})$ as follows.

,

• $\Sigma=\Sigma_{1}\cup\Sigma_{2}\cup\{S,x,y\}$

• $\Delta=\Delta_{1}\cup\Delta_{2}\cup\{x,y\}$

• $\Gamma=\Gamma_{1}\cup\Gamma_{2}$

• $\Omega=\Omega_{1}\cup\Omega_{2}$

• $P={P^{\prime}}_{1}\cup{P^{\prime}}_{2}\cup\{p_{01}\}$

• $R(P)=p_{01}{R^{\prime}}_{1}({P^{\prime}}_{1}){R^{\prime}}_{2}({P^{\prime}}_{2})$

It is clear that the underlying graph obtained after chain concatenation of graphs $G_{1}$ and $G_{2}$ is identical to the the underlying graph $G$ which is the result of this construction. ∎

Consider an $nc-eNCE$ graph grammar $ncG_{1}=(\Sigma_{1},\Delta_{1},\Gamma_{1},\Omega_{1},P_{1},S_{1},$ $R_{1}(P_{1}))$ generating the graph language $L_{1}$. We define a new $nc-eNCE$ graph grammar $ncG$ capable of generating the language $L=L_{1}^{\circledast}$ as follows.

,

• $\Sigma=\Sigma_{1}\cup\{S,x,c\}$

• $\Delta=\Delta_{1}\cup\{x,c\}$

• $\Gamma=\Gamma_{1}$

• $\Omega=\Omega_{1}$

• $P={P^{\prime}}_{1}\cup\{p_{01},p_{02}\}$

• $R(P)=(p_{01}.{R^{\prime}}_{1}({P^{\prime}}_{1}))^{*}.p_{02}$

Initially the production $p_{01}$ replaces the start node $S$ of $ncG$ with a copy of the start graph $S_{1}$ of $ncG_{1}$ and the node $S$ itself. Now ${R^{\prime}}_{1}({P^{\prime}}_{1})$ replaces $S_{1}$ by a graph $G_{1}\in L_{1}$. The rules $p_{01}$ and ${R^{\prime}}_{1}({P^{\prime}}_{1})$ are recursively applied to further generate new components of the resultant graph. Finally the derivation terminates when $p_{02}$ is applied. Hence it is clear that $R(P)$ generates $L_{1}^{\circledast}$. ∎